Combinatorics in Banach space theory Lecture 12

Size: px
Start display at page:

Download "Combinatorics in Banach space theory Lecture 12"

Transcription

1 Combinatorics in Banach space theory Lecture The next lemma considerably strengthens the assertion of Lemma.6(b). Lemma.9. For every Banach space X and any n N, either all the numbers n b n (X), c n (X) and d n (X) are equal to, or they are all less than. Proof. In view of Lemma.6(b), we shall only prove that c n = implies d n = and d n = implies n b n =, for every n N. First, assume c n = and fix any δ > 0. Then, there exist x,..., x n X such that n j= x j = n and r j (t)x j dt ( δ) n. (.) 0 j= To show that d n = we need to give an upper estimate for max j x j. Namely, we will show that max x j + (δn) /. (.) j n To this end, choose k 0 [n] such that x k0 = max j x j and apply Minkowski s inequality to get ( ) / ( ( n / x k0 = x k0 xk0 x j ) ) / ( ) /. + x j (.3) j= Moreover, by (.), we have ( xk0 x j ) j= i,j n j= j= ( xi x j ) ( = n x j x j j= ( = n x j ) n j= n ( δ) n 4δn, 0 j= r j (t)x j dt which, jointly with (.3), yields the claimed inequality (.). Combining this inequality with (.) we get ( / r j (t)x j dt) 0 j= j= δ n max + (δn) x / j, j n ) whence d n δ. + (δn) / δ 0

2 In the second part of the proof, we assume d n = and we want to show that n b n =. By the definition of d n, we have sup (x j ) n j= B X n ε j =± ε j x j = n. (.4) For any sequence (x j ) n j= B X pick a sequence (ˆε j ) n j= {, } n which minimises the norm of j ˆε jx j, that is, j ˆε jx j b n. Then, we have ε j =± j= ε n j x j = ˆε n j x j + n j= j= ε j =± (ε j ) (ˆε j ) ε j x j ( b n n + ( n )n ). Taking the supremum over all sequences (x j ) n j= B X, and making use of (.4), we obtain n n (b n + ( n )n ) which forces b n to be equal to n and completes the proof. Proof of Theorem.3. The equivalence (i) (ii) follows directly from Lemmas.4 and.5. Hence, both (i) and (ii) are in turn equivalent to (iii), in view of Lemma.9. Now, assuming (iii) choose k N so that σ := c k <. By the submultiplicativity of (c n ) n= (Lemma.6(d)), we have c k r σ r for each r N. For an arbitrary n N we may write n = k r + s with some r N 0 and 0 s < k r+ k r. Then, by Lemma.6(a), we have j= c n c k r+ kr + s +... k r + s k r+ k k r+ = r+ k k r + s r+ = k, k r whence c n k σ r+ for each n N, which implies (iv). We have proved so far that all conditions (i)-(iv) are pairwise equivalent. Now, assume that any of the conditions (i)-(iv) is satisfied (then, all of them are satisfied). We will show that if X contains l n s uniformly and λ is the uniformity constant, then b n λ n for every n N which will give a contradiction. So, suppose there is a sequence (E n ) n= of finite-dimensional subspaces of X such that for every n N there exists an isomorphism U n : l n E n such that U n = and Un λ. Fix any n N and let (ε j ) n j= {, } n be a sequence of signs that minimises the norm of n j= ε ju n (e j ). Then, n = j= ε j e j l n U n ε j U n (e j ) λ ε j U n (e j ) = λb n X X j= and so the proof of the implication (i) (v) has been completed. The implication (v) (vi) is trivial. Finally, to complete the proof we shall show that (vi) (i). Again, we prove it by contraposition. So, suppose that X is not B-convex. In light of Lemma.7, to guarantee l being finitely representable in X we shall merely show that X contains all l n s λ-uniformly, for every λ > (i.e. for every n N and λ > there exists an n- dimensional subspace E of X such that d BM (E, l n ) λ). j=

3 Fix any n N and λ >, and set δ = λ (0, ). Since X is not B-convex, there exist x,..., x n B X such that ε j x j n δ for every (ε j ) n j= {, } n. (.5) j= As shall be expected, the vectors x,..., x n span an almost isometric copy of l n inside X. In order to show this, we shall prove that the inequality λ a j a j x j a j (.6) j= j= holds true for arbitrary scalars a,..., a n. In fact, this would mean that the sequence (x j ) n j= is equivalent to the canonical basis (e j ) n j= of l n in the sense that there is an isomorphism T : span{x j } n j= l n satisfying T x j = e j for each j [n]. Moreover, the estimates (.6) would also indicate that T λ and T, thus d BM (span{x j } n j=, l n ) λ, as desired. After these explanations we may proceed to the proof of inequalities (.6). The second inequality in (.6) is obvious, since x j B X for each j [n]. For proving the first one, we may assume with no loss of generality that j a j =. Then, in view of (.5), we have ( n δ sgn(a j )x j = sgn(aj ) ( a j ) ) x j j= j= ( a j ) + a j x j = n + a j x j, j= which gives the first inequality in (.6). The proof is completed. Now, when we understand the essence of being B-convex pretty well, we shall quote the main result from Beck s paper [Bec6]. We will not prove it here, because it is beyond the main scope of our interest. However, it must be recalled, since it is a pioneering result which initiated the study of B-convex spaces. Recall that if (Ω, Σ, µ) is a measure space and X is a Banach space, then a map f : Ω X is called µ-measurable (or µ-strongly measurable), provided that there exists a sequence (g n ) n= of Σ-measurable step functions such that lim n f g n = 0 holds true µ-almost everywhere. The map f is Bochner integrable if and only if f dµ < ; Ω see [DU77, II.]. Now, if (Ω, Σ, P) is a probabilistic space, then any P-measurable function ξ : Ω X is called a random variable. If it is Bochner integrable, then the integral ξ dp Ω is called its expectation value and is denoted as Eξ. We also define the variation D ξ of the random variable ξ by D ξ = ξ Ω Eξ dp. We say that ξ is symmetric, if there exists a measure preserving map φ: Ω Ω such that ξ φ = ξ. A sequence (ξ n ) n= of X-valued random variables is called independent, whenever for every finite sequence By the classical Pettis measurability theorem, a function f : Ω X is µ-measurable if and only if it is essentially separably valued (i.e. for some µ-measure zero set E Σ the range f(ω \ E) is a separable subset of X) and weakly µ-measurable (i.e. for every x X the scalar function x f is Σ-measurable); see [DU77, II.]. j= j= j= 3

4 n <... < n k of natural numbers, and every choice of Borel sets B,..., B k X, we have k P(ξ n B... ξ nk B k ) = P(ξ nj B j ). The main results of Beck s paper [Bec6] may be summarised in the following theorem which, roughly speaking, asserts that the strong law of large numbers for X-valued random variables is valid if and only if X is B-convex. Theorem.0 (Beck, 96). Let (Ω, Σ, P) be a probabilistic space and X be a Banach space. If X is B-convex, then for every independent sequence (ξ n ) n= of Bochner integrable random variables ξ n : Ω X satisfying Eξ n = 0 for n N and sup n D ξ n < we have n j= ξ j n j= 0 P-almost everywhere. Moreover, if X fails to be B-convex, then there exists an independent sequence (ξ n ) n= of Bochner integrable, symmetric random variables ξ n : Ω X satisfying Eξ n = 0 and ξ n almost everywhere (n N) and such that ess sup Ω lim sup n n ξ j =. At the end of this section, we derive another infinite-dimensional counterpart of the Lyapunov convexity theorem for vector measures taking values in B-convex Banach spaces. The proof is essentially the same as the proof of the quantitative Theorem.6; just instead of Steinitz s lemma we use the basic property of B-convex spaces. Theorem. (V. Kadets, 99). If µ: Σ X is a σ-additive, non-atomic vector measure of bounded variation, defined on a σ-algebra Σ and taking values in a B-convex Banach space X, then the closure of µ(σ) is convex. Proof. Of course, it is enough to show that co(µ(σ)) = 0. This will follow from Lemma.8, if we prove that for every A Σ and ε > 0 there exists A ε A, A ε Σ, such that µ(a ε ) µ(a) ε. So, fix any A Σ and ε > 0. Let M = µ (Ω) be the total variation of µ. In view of Proposition., applied to the variation of µ, there exists a partition {A,..., A n } Π(A) such that µ (A j ) ε for each j [n] and n is the least natural number for which nε µ (A). Hence, (n )ε < µ (A) M, so n < ε M + < ε M for ε small enough. Now, repeating the calculation from the proof of Theorem.6, with the notation A J = j J A j for J [n], we obtain min J [n] µ(a J) µ(a) = min ε j µ(a j ) ε j =± b n max µ(a j) j n b nε < Mn b n. j= By the characterisation of B-convexity (see Theorem.3), the last expression goes to zero as n, thus the proof is completed. j= 4

5 Remark.. It is by no means straightforward to state what precisely the relation is between Theorem. and Uhl s Theorem.0. While it is easy to see that the latter does not follow from the former (for instance, l is a separable dual space which is not B-convex), it appears to be quite difficult to show that there exists any B-convex Banach space which is not reflexive (if it was not true, then Theorem. would be a direct consequence of Theorem.0). In 964, after discovering that B-convexity and reflexivity have much in common, James conjectured that every B-convex space is necessarily reflexive. Some partial confirmation of this conjecture was given in terms of (k, ε)-convexity. We say that a Banach space X is (k, ε)-convex (for some k and ε > 0), if sup (x j ) k j= B X min ε j =± k ε j x j ( ε)k. j= James [Jam64] showed that (, ε)-convexity implies reflexivity, for every ε > 0. In 973, Giesy confirmed James conjecture for Banach lattices ([Gi73]) and then he proved that for every k 3 and ε > 9 k every (k, ε)-convex Banach space must be reflexive. Finally, 4 in 974, James [Jam74] himself disproved his conjecture by constructing a non-reflexive Banach space X J such that for some λ > there is no isomorphism T from l 3 into X J satisfying λ x T x λ x for x l3 (hence, l is not finitely representable in X J and therefore X J is B-convex). He improved his construction later in [Jam78]. Consequently, Theorems.0 and. are incomparable. 3 B-convexity is a 3SP property In Section 8 we described the general framework for the three-space problem. Now, we turn our attention to the question whether B-convexity is a 3SP property, to which an affirmative answer, in the category of Banach spaces, was obtained by Giesy [Gie66]. Precisely saying, if X, Y and Z are all Banach spaces which form a short exact sequence 0 Y Z X 0 and if X and Y are B-convex, then Z also must be B-convex. Giesy s result was later expanded by Kalton [Kal78] who showed that it is not essential to assume that Z is a Banach space (which is, in fact, equivalent to assuming that Z is locally convex; see Proposition 3. below and recall the classical result, e.g. [Rud9, Theorem.39], which says that a linear topological space is normed iff it is locally bounded and locally convex). Instead, we may merely assume that Z is a locally bounded F -space (that is, a linear topological space having a bounded zero neighbourhood and being completely metrisable). Therefore, for the time being, we move to the world of linear topological spaces (not necessarily locally convex) and we aim to prove in this section the Kalton Giesy theorem, in particular to show how local convexity of Z follows from B-convexity of X and Y in the exact sequence above. Let us start with a simple result, due to Roelcke and Dierolf [RD8], which justifies our interest in locally bounded F -spaces. 5

6 Proposition 3. (Roelcke, Dierolf, 98). Let X be a linear topological space and Y be a closed subspace of X. (a) If both Y and X/Y are F -spaces, then so is X. (b) If both Y and X/Y are locally bounded, then so is X. Proof. (a): Recall that a linear topological space Z is metrisable if and only if it has a countable basis of zero neighbourhoods and in such a case there exists an invariant metric d on Z (i.e. d(x + z, y + z) = d(x, y) for all x, y, z Z) which is consistent with the given topology on Z (see [Rud9, Theorem.4]). So, there exists a decreasing sequence (V n ) n= of open subsets of X such that (Y V n ) n= is a basis of zero neighbourhoods in Y and (π(v n )) n= is a basis of zero neighbourhoods in X/Y, where π : X X/Y is the canonical map. Fix any zero neighbourhood U X; we are to prove that V m U for some m N. Pick any zero neighbourhood W X with W + W U. There exist m, n N with m > n such that Y V n W and π(v m ) π(w V n+ ), hence V m (W V n+ ) + Y. Fix any v V m ; v = w + y for some w W V n+ and y Y. Then and, consequently, y = v w Y [V m (W V n+ )] Y (V m V n+ ) V m (W V n+ ) [Y (V m V n+ )] (W V n+ ) + [Y (V n+ V n+ )] (W V n+ ) + (Y V n ) W + W U. Therefore, (V n ) n= is a (countable) basis of zero neighbourhoods in X. Now, assume that Y and X/Y are F -spaces and let d X be an invariant metric on X. Then, the formula d X/Y (π(x), π(y)) = inf{d X (x y, z): z Y } (3.) defines an invariant metric on X/Y which is consistent with the quotient topology on X/Y (see [Rud9, Theorem.4]). It follows that if (x n ) n= is a Cauchy sequence in X, then (π(x n )) n= is a Cauchy sequence in X/Y. Hence, there exists ξ X/Y such that lim n d X/Y (π(x n ), ξ) = 0. Choose any x 0 X with ξ = π(x 0 ). Then, in view of formula (3.), there exists a sequence (z n ) n= Y satisfying lim n d X (x n x 0, z n ) = 0. The inequality d X (z n, z m ) d X (x n x 0, z n ) + d X (x m x 0, z m ) + d X (x n, x m ) shows that (z n ) n= is a Cauchy sequence in Y and hence it is convergent to some z 0 Y. Then lim n d X (x n x 0, z 0 ) = 0, thus the sequence (x n ) n= converges to x 0 + z 0. (b): Assume that both Y and X/Y are locally bounded. Then, there is a zero neighbourhood U X such that both π(u) and Y U are bounded. Let V X be a balanced zero neighbourhood such that V + V U. We claim that V is bounded. Assume not and let (x n ) n= be a sequence in V such that ( n x n) n= is unbounded. Then n π(x n) 0 and hence n x n + y n 0 for some sequence (y n ) n= Y. Therefore, for n s large enough we have n x n + y n V, so y n V n V V + V U, which means that (y n) n= is bounded and forces the sequence ( n x n) n= to be bounded as well; a contradiction. 6

7 To prepare for the proof of the Kalton Giesy theorem we need to recall some material on geometric properties of locally bounded spaces. Definition 3.. Let X be a real or complex vector space. By a quasi-norm we mean any function X x x that satisfies the following three conditions: (i) x > 0 for every x X, x 0; (ii) λx = λ x for every x X and every scalar λ; (iii) x + y k ( x + y ) for all x, y X, where k is a constant independent of x and y. Any such constant is called the modulus of concavity of X. The class of quasi-normed spaces and the class of locally bounded spaces coincide. In fact, if X is quasi-normed by a quasi-norm, then the collection {εb X : ε > 0}, where B X = {x X : x }, is a base of bounded zero neighbourhoods for some linear topology on X. Conversely, if X is a locally bounded linear topological space, then the Minkowski functional of any bounded, balanced zero neighbourhood defines a quasi-norm on X which is consistent with the original topology. For quasi-normed spaces the notions of equivalent norms, quotient space, operator norm etc. are defined in a very the same way as for normed spaces. Definition 3.3. Let X be a quasi-normed space and 0 < p. We say that X is locally p-convex, provided that there exists a quasi-norm on X, equivalent to the original one, such that x + y p x p + y p for all x, y X. In such a case, the quasi-norm is called a p-norm. Let us now quote the fundamental result proved independently by Aoki and Rolewicz. The proof of the version presented below is explained in details in the hint to [Gra08, Exercise.4.6]. Theorem 3.4 (Aoki, 94 & Rolewicz, 957). Let be a quasi-norm on a vector space X with modulus of concavity k and let 0 < p be the solution of (k) p =. Then, there exists a p-norm on X which is equivalent to. More precisely, the quasi-norm satisfies the inequality x x n p 4 ( x p +... x n p) for all x,..., x n X and the quasi-norm given by the formula { ( ) /p x = inf x j p : x,..., x n X and x = j= } x j j= is a p-norm equivalent to. Let X be a quasi-normed space and be a quasi-norm on X. Certainly, all the formulas defining the quantities: a n (X), b n (X), c n (X) and d n (X) (see Section ) make sense in this more general setting. We are ready to make the first essential step towards the proof of the Kalton Giesy theorem. Proposition 3.5 (Kalton, 978). A quasi-normed space X is locally convex if and only if sup n n a n (X) <. 7

8 Proof. The only if part is obvious. For the if part the only thing to be proved is that the convex hull of the unit ball {x X : x } is bounded. So, assume there is a constant C < such that a n Cn for each n N and fix arbitrary vectors x,..., x n X with x j (for j [n]) and arbitrary non-negative numbers α,..., α n with n j= α j =. For every j [n] and m N set k j,m = mα j. Plainly, lim m m k j,m = α j. For every m N we have k j,m x j a n j= k C k j,m j,m, hence j= j= m x j C. Letting m, and denoting k the modulus of concavity, we arrive at ( ) k j,m α j x j k m x j + ( α j k ) j,m x j m j= j= j= ( k C + k α j k ) j,m x j m k C, m so the result follows. k j,m j= j= 8

PROBLEMS. (b) (Polarization Identity) Show that in any inner product space

PROBLEMS. (b) (Polarization Identity) Show that in any inner product space 1 Professor Carl Cowen Math 54600 Fall 09 PROBLEMS 1. (Geometry in Inner Product Spaces) (a) (Parallelogram Law) Show that in any inner product space x + y 2 + x y 2 = 2( x 2 + y 2 ). (b) (Polarization

More information

Problem Set 6: Solutions Math 201A: Fall a n x n,

Problem Set 6: Solutions Math 201A: Fall a n x n, Problem Set 6: Solutions Math 201A: Fall 2016 Problem 1. Is (x n ) n=0 a Schauder basis of C([0, 1])? No. If f(x) = a n x n, n=0 where the series converges uniformly on [0, 1], then f has a power series

More information

Overview of normed linear spaces

Overview of normed linear spaces 20 Chapter 2 Overview of normed linear spaces Starting from this chapter, we begin examining linear spaces with at least one extra structure (topology or geometry). We assume linearity; this is a natural

More information

RESTRICTED UNIFORM BOUNDEDNESS IN BANACH SPACES

RESTRICTED UNIFORM BOUNDEDNESS IN BANACH SPACES RESTRICTED UNIFORM BOUNDEDNESS IN BANACH SPACES OLAV NYGAARD AND MÄRT PÕLDVERE Abstract. Precise conditions for a subset A of a Banach space X are known in order that pointwise bounded on A sequences of

More information

Examples of Dual Spaces from Measure Theory

Examples of Dual Spaces from Measure Theory Chapter 9 Examples of Dual Spaces from Measure Theory We have seen that L (, A, µ) is a Banach space for any measure space (, A, µ). We will extend that concept in the following section to identify an

More information

Lecture 4 Lebesgue spaces and inequalities

Lecture 4 Lebesgue spaces and inequalities Lecture 4: Lebesgue spaces and inequalities 1 of 10 Course: Theory of Probability I Term: Fall 2013 Instructor: Gordan Zitkovic Lecture 4 Lebesgue spaces and inequalities Lebesgue spaces We have seen how

More information

Exercise Solutions to Functional Analysis

Exercise Solutions to Functional Analysis Exercise Solutions to Functional Analysis Note: References refer to M. Schechter, Principles of Functional Analysis Exersize that. Let φ,..., φ n be an orthonormal set in a Hilbert space H. Show n f n

More information

MAT 771 FUNCTIONAL ANALYSIS HOMEWORK 3. (1) Let V be the vector space of all bounded or unbounded sequences of complex numbers.

MAT 771 FUNCTIONAL ANALYSIS HOMEWORK 3. (1) Let V be the vector space of all bounded or unbounded sequences of complex numbers. MAT 771 FUNCTIONAL ANALYSIS HOMEWORK 3 (1) Let V be the vector space of all bounded or unbounded sequences of complex numbers. (a) Define d : V V + {0} by d(x, y) = 1 ξ j η j 2 j 1 + ξ j η j. Show that

More information

ON WEAK INTEGRABILITY AND BOUNDEDNESS IN BANACH SPACES

ON WEAK INTEGRABILITY AND BOUNDEDNESS IN BANACH SPACES ON WEAK INTEGRABILITY AND BOUNDEDNESS IN BANACH SPACES TROND A. ABRAHAMSEN, OLAV NYGAARD, AND MÄRT PÕLDVERE Abstract. Thin and thick sets in normed spaces were defined and studied by M. I. Kadets and V.

More information

Functional Analysis. Martin Brokate. 1 Normed Spaces 2. 2 Hilbert Spaces The Principle of Uniform Boundedness 32

Functional Analysis. Martin Brokate. 1 Normed Spaces 2. 2 Hilbert Spaces The Principle of Uniform Boundedness 32 Functional Analysis Martin Brokate Contents 1 Normed Spaces 2 2 Hilbert Spaces 2 3 The Principle of Uniform Boundedness 32 4 Extension, Reflexivity, Separation 37 5 Compact subsets of C and L p 46 6 Weak

More information

CHAPTER V DUAL SPACES

CHAPTER V DUAL SPACES CHAPTER V DUAL SPACES DEFINITION Let (X, T ) be a (real) locally convex topological vector space. By the dual space X, or (X, T ), of X we mean the set of all continuous linear functionals on X. By the

More information

4 Hilbert spaces. The proof of the Hilbert basis theorem is not mathematics, it is theology. Camille Jordan

4 Hilbert spaces. The proof of the Hilbert basis theorem is not mathematics, it is theology. Camille Jordan The proof of the Hilbert basis theorem is not mathematics, it is theology. Camille Jordan Wir müssen wissen, wir werden wissen. David Hilbert We now continue to study a special class of Banach spaces,

More information

5 Measure theory II. (or. lim. Prove the proposition. 5. For fixed F A and φ M define the restriction of φ on F by writing.

5 Measure theory II. (or. lim. Prove the proposition. 5. For fixed F A and φ M define the restriction of φ on F by writing. 5 Measure theory II 1. Charges (signed measures). Let (Ω, A) be a σ -algebra. A map φ: A R is called a charge, (or signed measure or σ -additive set function) if φ = φ(a j ) (5.1) A j for any disjoint

More information

Functional Analysis HW #1

Functional Analysis HW #1 Functional Analysis HW #1 Sangchul Lee October 9, 2015 1 Solutions Solution of #1.1. Suppose that X

More information

The small ball property in Banach spaces (quantitative results)

The small ball property in Banach spaces (quantitative results) The small ball property in Banach spaces (quantitative results) Ehrhard Behrends Abstract A metric space (M, d) is said to have the small ball property (sbp) if for every ε 0 > 0 there exists a sequence

More information

Spectral theory for compact operators on Banach spaces

Spectral theory for compact operators on Banach spaces 68 Chapter 9 Spectral theory for compact operators on Banach spaces Recall that a subset S of a metric space X is precompact if its closure is compact, or equivalently every sequence contains a Cauchy

More information

FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES. 1. Compact Sets

FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES. 1. Compact Sets FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES CHRISTOPHER HEIL 1. Compact Sets Definition 1.1 (Compact and Totally Bounded Sets). Let X be a metric space, and let E X be

More information

Weak Topologies, Reflexivity, Adjoint operators

Weak Topologies, Reflexivity, Adjoint operators Chapter 2 Weak Topologies, Reflexivity, Adjoint operators 2.1 Topological vector spaces and locally convex spaces Definition 2.1.1. [Topological Vector Spaces and Locally convex Spaces] Let E be a vector

More information

Boundedly complete weak-cauchy basic sequences in Banach spaces with the PCP

Boundedly complete weak-cauchy basic sequences in Banach spaces with the PCP Journal of Functional Analysis 253 (2007) 772 781 www.elsevier.com/locate/jfa Note Boundedly complete weak-cauchy basic sequences in Banach spaces with the PCP Haskell Rosenthal Department of Mathematics,

More information

MAT 578 FUNCTIONAL ANALYSIS EXERCISES

MAT 578 FUNCTIONAL ANALYSIS EXERCISES MAT 578 FUNCTIONAL ANALYSIS EXERCISES JOHN QUIGG Exercise 1. Prove that if A is bounded in a topological vector space, then for every neighborhood V of 0 there exists c > 0 such that tv A for all t > c.

More information

3 (Due ). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?

3 (Due ). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure? MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due ). Show that the open disk x 2 + y 2 < 1 is a countable union of planar elementary sets. Show that the closed disk x 2 + y 2 1 is a countable

More information

Compact operators on Banach spaces

Compact operators on Banach spaces Compact operators on Banach spaces Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto November 12, 2017 1 Introduction In this note I prove several things about compact

More information

l(y j ) = 0 for all y j (1)

l(y j ) = 0 for all y j (1) Problem 1. The closed linear span of a subset {y j } of a normed vector space is defined as the intersection of all closed subspaces containing all y j and thus the smallest such subspace. 1 Show that

More information

2 (Bonus). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?

2 (Bonus). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure? MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due 9/5). Prove that every countable set A is measurable and µ(a) = 0. 2 (Bonus). Let A consist of points (x, y) such that either x or y is

More information

(convex combination!). Use convexity of f and multiply by the common denominator to get. Interchanging the role of x and y, we obtain that f is ( 2M ε

(convex combination!). Use convexity of f and multiply by the common denominator to get. Interchanging the role of x and y, we obtain that f is ( 2M ε 1. Continuity of convex functions in normed spaces In this chapter, we consider continuity properties of real-valued convex functions defined on open convex sets in normed spaces. Recall that every infinitedimensional

More information

Introduction to Real Analysis Alternative Chapter 1

Introduction to Real Analysis Alternative Chapter 1 Christopher Heil Introduction to Real Analysis Alternative Chapter 1 A Primer on Norms and Banach Spaces Last Updated: March 10, 2018 c 2018 by Christopher Heil Chapter 1 A Primer on Norms and Banach Spaces

More information

Banach Spaces II: Elementary Banach Space Theory

Banach Spaces II: Elementary Banach Space Theory BS II c Gabriel Nagy Banach Spaces II: Elementary Banach Space Theory Notes from the Functional Analysis Course (Fall 07 - Spring 08) In this section we introduce Banach spaces and examine some of their

More information

Real Analysis Notes. Thomas Goller

Real Analysis Notes. Thomas Goller Real Analysis Notes Thomas Goller September 4, 2011 Contents 1 Abstract Measure Spaces 2 1.1 Basic Definitions........................... 2 1.2 Measurable Functions........................ 2 1.3 Integration..............................

More information

Measure Theory on Topological Spaces. Course: Prof. Tony Dorlas 2010 Typset: Cathal Ormond

Measure Theory on Topological Spaces. Course: Prof. Tony Dorlas 2010 Typset: Cathal Ormond Measure Theory on Topological Spaces Course: Prof. Tony Dorlas 2010 Typset: Cathal Ormond May 22, 2011 Contents 1 Introduction 2 1.1 The Riemann Integral........................................ 2 1.2 Measurable..............................................

More information

Extreme points of compact convex sets

Extreme points of compact convex sets Extreme points of compact convex sets In this chapter, we are going to show that compact convex sets are determined by a proper subset, the set of its extreme points. Let us start with the main definition.

More information

Functional Analysis Exercise Class

Functional Analysis Exercise Class Functional Analysis Exercise Class Week 2 November 6 November Deadline to hand in the homeworks: your exercise class on week 9 November 13 November Exercises (1) Let X be the following space of piecewise

More information

1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer 11(2) (1989),

1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer 11(2) (1989), Real Analysis 2, Math 651, Spring 2005 April 26, 2005 1 Real Analysis 2, Math 651, Spring 2005 Krzysztof Chris Ciesielski 1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer

More information

Best approximations in normed vector spaces

Best approximations in normed vector spaces Best approximations in normed vector spaces Mike de Vries 5699703 a thesis submitted to the Department of Mathematics at Utrecht University in partial fulfillment of the requirements for the degree of

More information

Functional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability...

Functional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability... Functional Analysis Franck Sueur 2018-2019 Contents 1 Metric spaces 1 1.1 Definitions........................................ 1 1.2 Completeness...................................... 3 1.3 Compactness......................................

More information

THEOREMS, ETC., FOR MATH 515

THEOREMS, ETC., FOR MATH 515 THEOREMS, ETC., FOR MATH 515 Proposition 1 (=comment on page 17). If A is an algebra, then any finite union or finite intersection of sets in A is also in A. Proposition 2 (=Proposition 1.1). For every

More information

Probability and Measure

Probability and Measure Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 84 Paper 4, Section II 26J Let (X, A) be a measurable space. Let T : X X be a measurable map, and µ a probability

More information

Notions such as convergent sequence and Cauchy sequence make sense for any metric space. Convergent Sequences are Cauchy

Notions such as convergent sequence and Cauchy sequence make sense for any metric space. Convergent Sequences are Cauchy Banach Spaces These notes provide an introduction to Banach spaces, which are complete normed vector spaces. For the purposes of these notes, all vector spaces are assumed to be over the real numbers.

More information

n [ F (b j ) F (a j ) ], n j=1(a j, b j ] E (4.1)

n [ F (b j ) F (a j ) ], n j=1(a j, b j ] E (4.1) 1.4. CONSTRUCTION OF LEBESGUE-STIELTJES MEASURES In this section we shall put to use the Carathéodory-Hahn theory, in order to construct measures with certain desirable properties first on the real line

More information

1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3

1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3 Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,

More information

Banach Spaces V: A Closer Look at the w- and the w -Topologies

Banach Spaces V: A Closer Look at the w- and the w -Topologies BS V c Gabriel Nagy Banach Spaces V: A Closer Look at the w- and the w -Topologies Notes from the Functional Analysis Course (Fall 07 - Spring 08) In this section we discuss two important, but highly non-trivial,

More information

Functional Analysis Exercise Class

Functional Analysis Exercise Class Functional Analysis Exercise Class Wee November 30 Dec 4: Deadline to hand in the homewor: your exercise class on wee December 7 11 Exercises with solutions Recall that every normed space X can be isometrically

More information

NOTES ON VECTOR-VALUED INTEGRATION MATH 581, SPRING 2017

NOTES ON VECTOR-VALUED INTEGRATION MATH 581, SPRING 2017 NOTES ON VECTOR-VALUED INTEGRATION MATH 58, SPRING 207 Throughout, X will denote a Banach space. Definition 0.. Let ϕ(s) : X be a continuous function from a compact Jordan region R n to a Banach space

More information

Functional Analysis HW #5

Functional Analysis HW #5 Functional Analysis HW #5 Sangchul Lee October 29, 2015 Contents 1 Solutions........................................ 1 1 Solutions Exercise 3.4. Show that C([0, 1]) is not a Hilbert space, that is, there

More information

Course Notes for Functional Analysis I, Math , Fall Th. Schlumprecht

Course Notes for Functional Analysis I, Math , Fall Th. Schlumprecht Course Notes for Functional Analysis I, Math 655-601, Fall 2011 Th. Schlumprecht December 13, 2011 2 Contents 1 Some Basic Background 5 1.1 Normed Linear Spaces, Banach Spaces............. 5 1.2 Operators

More information

2. Dual space is essential for the concept of gradient which, in turn, leads to the variational analysis of Lagrange multipliers.

2. Dual space is essential for the concept of gradient which, in turn, leads to the variational analysis of Lagrange multipliers. Chapter 3 Duality in Banach Space Modern optimization theory largely centers around the interplay of a normed vector space and its corresponding dual. The notion of duality is important for the following

More information

Topological vectorspaces

Topological vectorspaces (July 25, 2011) Topological vectorspaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ Natural non-fréchet spaces Topological vector spaces Quotients and linear maps More topological

More information

Measurable functions are approximately nice, even if look terrible.

Measurable functions are approximately nice, even if look terrible. Tel Aviv University, 2015 Functions of real variables 74 7 Approximation 7a A terrible integrable function........... 74 7b Approximation of sets................ 76 7c Approximation of functions............

More information

Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University

Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University February 7, 2007 2 Contents 1 Metric Spaces 1 1.1 Basic definitions...........................

More information

Recall that if X is a compact metric space, C(X), the space of continuous (real-valued) functions on X, is a Banach space with the norm

Recall that if X is a compact metric space, C(X), the space of continuous (real-valued) functions on X, is a Banach space with the norm Chapter 13 Radon Measures Recall that if X is a compact metric space, C(X), the space of continuous (real-valued) functions on X, is a Banach space with the norm (13.1) f = sup x X f(x). We want to identify

More information

REAL AND COMPLEX ANALYSIS

REAL AND COMPLEX ANALYSIS REAL AND COMPLE ANALYSIS Third Edition Walter Rudin Professor of Mathematics University of Wisconsin, Madison Version 1.1 No rights reserved. Any part of this work can be reproduced or transmitted in any

More information

Integration on Measure Spaces

Integration on Measure Spaces Chapter 3 Integration on Measure Spaces In this chapter we introduce the general notion of a measure on a space X, define the class of measurable functions, and define the integral, first on a class of

More information

Spectral Theory, with an Introduction to Operator Means. William L. Green

Spectral Theory, with an Introduction to Operator Means. William L. Green Spectral Theory, with an Introduction to Operator Means William L. Green January 30, 2008 Contents Introduction............................... 1 Hilbert Space.............................. 4 Linear Maps

More information

FUNCTIONAL ANALYSIS HAHN-BANACH THEOREM. F (m 2 ) + α m 2 + x 0

FUNCTIONAL ANALYSIS HAHN-BANACH THEOREM. F (m 2 ) + α m 2 + x 0 FUNCTIONAL ANALYSIS HAHN-BANACH THEOREM If M is a linear subspace of a normal linear space X and if F is a bounded linear functional on M then F can be extended to M + [x 0 ] without changing its norm.

More information

INTRODUCTION TO MEASURE THEORY AND LEBESGUE INTEGRATION

INTRODUCTION TO MEASURE THEORY AND LEBESGUE INTEGRATION 1 INTRODUCTION TO MEASURE THEORY AND LEBESGUE INTEGRATION Eduard EMELYANOV Ankara TURKEY 2007 2 FOREWORD This book grew out of a one-semester course for graduate students that the author have taught at

More information

Chapter 1. Measure Spaces. 1.1 Algebras and σ algebras of sets Notation and preliminaries

Chapter 1. Measure Spaces. 1.1 Algebras and σ algebras of sets Notation and preliminaries Chapter 1 Measure Spaces 1.1 Algebras and σ algebras of sets 1.1.1 Notation and preliminaries We shall denote by X a nonempty set, by P(X) the set of all parts (i.e., subsets) of X, and by the empty set.

More information

Integral Jensen inequality

Integral Jensen inequality Integral Jensen inequality Let us consider a convex set R d, and a convex function f : (, + ]. For any x,..., x n and λ,..., λ n with n λ i =, we have () f( n λ ix i ) n λ if(x i ). For a R d, let δ a

More information

FUNCTIONAL ANALYSIS CHRISTIAN REMLING

FUNCTIONAL ANALYSIS CHRISTIAN REMLING FUNCTIONAL ANALYSIS CHRISTIAN REMLING Contents 1. Metric and topological spaces 2 2. Banach spaces 12 3. Consequences of Baire s Theorem 30 4. Dual spaces and weak topologies 34 5. Hilbert spaces 50 6.

More information

Continuity of convolution and SIN groups

Continuity of convolution and SIN groups Continuity of convolution and SIN groups Jan Pachl and Juris Steprāns June 5, 2016 Abstract Let the measure algebra of a topological group G be equipped with the topology of uniform convergence on bounded

More information

NOTES ON FRAMES. Damir Bakić University of Zagreb. June 6, 2017

NOTES ON FRAMES. Damir Bakić University of Zagreb. June 6, 2017 NOTES ON FRAMES Damir Bakić University of Zagreb June 6, 017 Contents 1 Unconditional convergence, Riesz bases, and Bessel sequences 1 1.1 Unconditional convergence of series in Banach spaces...............

More information

i=1 β i,i.e. = β 1 x β x β 1 1 xβ d

i=1 β i,i.e. = β 1 x β x β 1 1 xβ d 66 2. Every family of seminorms on a vector space containing a norm induces ahausdorff locally convex topology. 3. Given an open subset Ω of R d with the euclidean topology, the space C(Ω) of real valued

More information

16 1 Basic Facts from Functional Analysis and Banach Lattices

16 1 Basic Facts from Functional Analysis and Banach Lattices 16 1 Basic Facts from Functional Analysis and Banach Lattices 1.2.3 Banach Steinhaus Theorem Another fundamental theorem of functional analysis is the Banach Steinhaus theorem, or the Uniform Boundedness

More information

HAAR NEGLIGIBILITY OF POSITIVE CONES IN BANACH SPACES

HAAR NEGLIGIBILITY OF POSITIVE CONES IN BANACH SPACES HAAR NEGLIGIBILITY OF POSITIVE CONES IN BANACH SPACES JEAN ESTERLE, ÉTIENNE MATHERON, AND PIERRE MOREAU Abstract. We discuss the Haar negligibility of the positive cone associated with a basic sequence

More information

MATHS 730 FC Lecture Notes March 5, Introduction

MATHS 730 FC Lecture Notes March 5, Introduction 1 INTRODUCTION MATHS 730 FC Lecture Notes March 5, 2014 1 Introduction Definition. If A, B are sets and there exists a bijection A B, they have the same cardinality, which we write as A, #A. If there exists

More information

Math 209B Homework 2

Math 209B Homework 2 Math 29B Homework 2 Edward Burkard Note: All vector spaces are over the field F = R or C 4.6. Two Compactness Theorems. 4. Point Set Topology Exercise 6 The product of countably many sequentally compact

More information

Normed Vector Spaces and Double Duals

Normed Vector Spaces and Double Duals Normed Vector Spaces and Double Duals Mathematics 481/525 In this note we look at a number of infinite-dimensional R-vector spaces that arise in analysis, and we consider their dual and double dual spaces

More information

A SHORT INTRODUCTION TO BANACH LATTICES AND

A SHORT INTRODUCTION TO BANACH LATTICES AND CHAPTER A SHORT INTRODUCTION TO BANACH LATTICES AND POSITIVE OPERATORS In tis capter we give a brief introduction to Banac lattices and positive operators. Most results of tis capter can be found, e.g.,

More information

6 Classical dualities and reflexivity

6 Classical dualities and reflexivity 6 Classical dualities and reflexivity 1. Classical dualities. Let (Ω, A, µ) be a measure space. We will describe the duals for the Banach spaces L p (Ω). First, notice that any f L p, 1 p, generates the

More information

Real Analysis, 2nd Edition, G.B.Folland Elements of Functional Analysis

Real Analysis, 2nd Edition, G.B.Folland Elements of Functional Analysis Real Analysis, 2nd Edition, G.B.Folland Chapter 5 Elements of Functional Analysis Yung-Hsiang Huang 5.1 Normed Vector Spaces 1. Note for any x, y X and a, b K, x+y x + y and by ax b y x + b a x. 2. It

More information

An Asymptotic Property of Schachermayer s Space under Renorming

An Asymptotic Property of Schachermayer s Space under Renorming Journal of Mathematical Analysis and Applications 50, 670 680 000) doi:10.1006/jmaa.000.7104, available online at http://www.idealibrary.com on An Asymptotic Property of Schachermayer s Space under Renorming

More information

I teach myself... Hilbert spaces

I teach myself... Hilbert spaces I teach myself... Hilbert spaces by F.J.Sayas, for MATH 806 November 4, 2015 This document will be growing with the semester. Every in red is for you to justify. Even if we start with the basic definition

More information

THE RADON NIKODYM PROPERTY AND LIPSCHITZ EMBEDDINGS

THE RADON NIKODYM PROPERTY AND LIPSCHITZ EMBEDDINGS THE RADON NIKODYM PROPERTY AND LIPSCHITZ EMBEDDINGS Motivation The idea here is simple. Suppose we have a Lipschitz homeomorphism f : X Y where X and Y are Banach spaces, namely c 1 x y f (x) f (y) c 2

More information

Hilbert spaces. 1. Cauchy-Schwarz-Bunyakowsky inequality

Hilbert spaces. 1. Cauchy-Schwarz-Bunyakowsky inequality (October 29, 2016) Hilbert spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/fun/notes 2016-17/03 hsp.pdf] Hilbert spaces are

More information

Bases in Banach spaces

Bases in Banach spaces Chapter 3 Bases in Banach spaces Like every vector space a Banach space X admits an algebraic or Hamel basis, i.e. a subset B X, so that every x X is in a unique way the (finite) linear combination of

More information

ON SUBSPACES OF NON-REFLEXIVE ORLICZ SPACES

ON SUBSPACES OF NON-REFLEXIVE ORLICZ SPACES ON SUBSPACES OF NON-REFLEXIVE ORLICZ SPACES J. ALEXOPOULOS May 28, 997 ABSTRACT. Kadec and Pelczýnski have shown that every non-reflexive subspace of L (µ) contains a copy of l complemented in L (µ). On

More information

Chapter 2 Metric Spaces

Chapter 2 Metric Spaces Chapter 2 Metric Spaces The purpose of this chapter is to present a summary of some basic properties of metric and topological spaces that play an important role in the main body of the book. 2.1 Metrics

More information

CHAPTER 7. Connectedness

CHAPTER 7. Connectedness CHAPTER 7 Connectedness 7.1. Connected topological spaces Definition 7.1. A topological space (X, T X ) is said to be connected if there is no continuous surjection f : X {0, 1} where the two point set

More information

Lecture Notes in Functional Analysis

Lecture Notes in Functional Analysis Lecture Notes in Functional Analysis Please report any mistakes, misprints or comments to aulikowska@mimuw.edu.pl LECTURE 1 Banach spaces 1.1. Introducton to Banach Spaces Definition 1.1. Let X be a K

More information

Maths 212: Homework Solutions

Maths 212: Homework Solutions Maths 212: Homework Solutions 1. The definition of A ensures that x π for all x A, so π is an upper bound of A. To show it is the least upper bound, suppose x < π and consider two cases. If x < 1, then

More information

Multi-normed spaces and multi-banach algebras. H. G. Dales. Leeds Semester

Multi-normed spaces and multi-banach algebras. H. G. Dales. Leeds Semester Multi-normed spaces and multi-banach algebras H. G. Dales Leeds Semester Leeds, 2 June 2010 1 Motivating problem Let G be a locally compact group, with group algebra L 1 (G). Theorem - B. E. Johnson, 1972

More information

Econ Lecture 3. Outline. 1. Metric Spaces and Normed Spaces 2. Convergence of Sequences in Metric Spaces 3. Sequences in R and R n

Econ Lecture 3. Outline. 1. Metric Spaces and Normed Spaces 2. Convergence of Sequences in Metric Spaces 3. Sequences in R and R n Econ 204 2011 Lecture 3 Outline 1. Metric Spaces and Normed Spaces 2. Convergence of Sequences in Metric Spaces 3. Sequences in R and R n 1 Metric Spaces and Metrics Generalize distance and length notions

More information

Tools from Lebesgue integration

Tools from Lebesgue integration Tools from Lebesgue integration E.P. van den Ban Fall 2005 Introduction In these notes we describe some of the basic tools from the theory of Lebesgue integration. Definitions and results will be given

More information

6. Duals of L p spaces

6. Duals of L p spaces 6 Duals of L p spaces This section deals with the problem if identifying the duals of L p spaces, p [1, ) There are essentially two cases of this problem: (i) p = 1; (ii) 1 < p < The major difference between

More information

Functional Analysis, Math 7320 Lecture Notes from August taken by Yaofeng Su

Functional Analysis, Math 7320 Lecture Notes from August taken by Yaofeng Su Functional Analysis, Math 7320 Lecture Notes from August 30 2016 taken by Yaofeng Su 1 Essentials of Topology 1.1 Continuity Next we recall a stronger notion of continuity: 1.1.1 Definition. Let (X, d

More information

LECTURE 15: COMPLETENESS AND CONVEXITY

LECTURE 15: COMPLETENESS AND CONVEXITY LECTURE 15: COMPLETENESS AND CONVEXITY 1. The Hopf-Rinow Theorem Recall that a Riemannian manifold (M, g) is called geodesically complete if the maximal defining interval of any geodesic is R. On the other

More information

Contents: 1. Minimization. 2. The theorem of Lions-Stampacchia for variational inequalities. 3. Γ -Convergence. 4. Duality mapping.

Contents: 1. Minimization. 2. The theorem of Lions-Stampacchia for variational inequalities. 3. Γ -Convergence. 4. Duality mapping. Minimization Contents: 1. Minimization. 2. The theorem of Lions-Stampacchia for variational inequalities. 3. Γ -Convergence. 4. Duality mapping. 1 Minimization A Topological Result. Let S be a topological

More information

MAA6617 COURSE NOTES SPRING 2014

MAA6617 COURSE NOTES SPRING 2014 MAA6617 COURSE NOTES SPRING 2014 19. Normed vector spaces Let X be a vector space over a field K (in this course we always have either K = R or K = C). Definition 19.1. A norm on X is a function : X K

More information

Part V. 17 Introduction: What are measures and why measurable sets. Lebesgue Integration Theory

Part V. 17 Introduction: What are measures and why measurable sets. Lebesgue Integration Theory Part V 7 Introduction: What are measures and why measurable sets Lebesgue Integration Theory Definition 7. (Preliminary). A measure on a set is a function :2 [ ] such that. () = 2. If { } = is a finite

More information

2.2 Annihilators, complemented subspaces

2.2 Annihilators, complemented subspaces 34CHAPTER 2. WEAK TOPOLOGIES, REFLEXIVITY, ADJOINT OPERATORS 2.2 Annihilators, complemented subspaces Definition 2.2.1. [Annihilators, Pre-Annihilators] Assume X is a Banach space. Let M X and N X. We

More information

Weak-Star Convergence of Convex Sets

Weak-Star Convergence of Convex Sets Journal of Convex Analysis Volume 13 (2006), No. 3+4, 711 719 Weak-Star Convergence of Convex Sets S. Fitzpatrick A. S. Lewis ORIE, Cornell University, Ithaca, NY 14853, USA aslewis@orie.cornell.edu www.orie.cornell.edu/

More information

MATH 426, TOPOLOGY. p 1.

MATH 426, TOPOLOGY. p 1. MATH 426, TOPOLOGY THE p-norms In this document we assume an extended real line, where is an element greater than all real numbers; the interval notation [1, ] will be used to mean [1, ) { }. 1. THE p

More information

Problem 1: Compactness (12 points, 2 points each)

Problem 1: Compactness (12 points, 2 points each) Final exam Selected Solutions APPM 5440 Fall 2014 Applied Analysis Date: Tuesday, Dec. 15 2014, 10:30 AM to 1 PM You may assume all vector spaces are over the real field unless otherwise specified. Your

More information

Translative Sets and Functions and their Applications to Risk Measure Theory and Nonlinear Separation

Translative Sets and Functions and their Applications to Risk Measure Theory and Nonlinear Separation Translative Sets and Functions and their Applications to Risk Measure Theory and Nonlinear Separation Andreas H. Hamel Abstract Recently defined concepts such as nonlinear separation functionals due to

More information

Characterisation of Accumulation Points. Convergence in Metric Spaces. Characterisation of Closed Sets. Characterisation of Closed Sets

Characterisation of Accumulation Points. Convergence in Metric Spaces. Characterisation of Closed Sets. Characterisation of Closed Sets Convergence in Metric Spaces Functional Analysis Lecture 3: Convergence and Continuity in Metric Spaces Bengt Ove Turesson September 4, 2016 Suppose that (X, d) is a metric space. A sequence (x n ) X is

More information

Continuity of convex functions in normed spaces

Continuity of convex functions in normed spaces Continuity of convex functions in normed spaces In this chapter, we consider continuity properties of real-valued convex functions defined on open convex sets in normed spaces. Recall that every infinitedimensional

More information

On the topology of pointwise convergence on the boundaries of L 1 -preduals. Warren B. Moors

On the topology of pointwise convergence on the boundaries of L 1 -preduals. Warren B. Moors On the topology of pointwise convergence on the boundaries of L 1 -preduals Warren B. Moors Department of Mathematics The University of Auckland Auckland New Zealand Dedicated to J ohn R. Giles Introduction

More information

3 Integration and Expectation

3 Integration and Expectation 3 Integration and Expectation 3.1 Construction of the Lebesgue Integral Let (, F, µ) be a measure space (not necessarily a probability space). Our objective will be to define the Lebesgue integral R fdµ

More information

CHAPTER 1. Metric Spaces. 1. Definition and examples

CHAPTER 1. Metric Spaces. 1. Definition and examples CHAPTER Metric Spaces. Definition and examples Metric spaces generalize and clarify the notion of distance in the real line. The definitions will provide us with a useful tool for more general applications

More information

An introduction to some aspects of functional analysis

An introduction to some aspects of functional analysis An introduction to some aspects of functional analysis Stephen Semmes Rice University Abstract These informal notes deal with some very basic objects in functional analysis, including norms and seminorms

More information

x n x or x = T -limx n, if every open neighbourhood U of x contains x n for all but finitely many values of n

x n x or x = T -limx n, if every open neighbourhood U of x contains x n for all but finitely many values of n Note: This document was written by Prof. Richard Haydon, a previous lecturer for this course. 0. Preliminaries This preliminary chapter is an attempt to bring together important results from earlier courses

More information

Lecture Notes. Functional Analysis in Applied Mathematics and Engineering. by Klaus Engel. University of L Aquila Faculty of Engineering

Lecture Notes. Functional Analysis in Applied Mathematics and Engineering. by Klaus Engel. University of L Aquila Faculty of Engineering Lecture Notes Functional Analysis in Applied Mathematics and Engineering by Klaus Engel University of L Aquila Faculty of Engineering 2012-2013 http://univaq.it/~engel ( = %7E) (Preliminary Version of

More information