# 2 Garrett: `A Good Spectral Theorem' 1. von Neumann algebras, density theorem The commutant of a subring S of a ring R is S 0 = fr 2 R : rs = sr; 8s 2

Save this PDF as:
Size: px
Start display at page:

Download "2 Garrett: `A Good Spectral Theorem' 1. von Neumann algebras, density theorem The commutant of a subring S of a ring R is S 0 = fr 2 R : rs = sr; 8s 2"

## Transcription

1 1 A Good Spectral Theorem c1996, Paul Garrett, version February 12, Measurable Hilbert bundles Measurable Banach bundles Direct integrals of Hilbert spaces Trivializing Hilbert bundles 1 Typeset with AMS-LaTeX

2 2 Garrett: `A Good Spectral Theorem' 1. von Neumann algebras, density theorem The commutant of a subring S of a ring R is S 0 = fr 2 R : rs = sr; 8s 2 Sg A von Neumann algebra is a -stable subalgebra A of the space B(V ) of bounded operators on a Hilbert space V so that A is closed in the strong topology on operators. Theorem: (von Neumann Density) Let A be a -stable subalgebra of the space B(V ) of bounded operators on a Hilbert space, and suppose that A contains the identity. Then A 00 = closure of A in the strong operator topology That is, if A is -closed and contains the scalar operators, then the strong-topology closure condition is equivalent to the purely algebraic condition A 00 = A Very often the von Neumann algebra in question will indeed include scalar operators, but we reserve the right to have it be otherwise. 2. Direct integrals of Hilbert spaces: denitions Let (; ) be a measure space. We assume that is a positive regular Borel measure. Further, we suppose that is -nite, i.e., is a countable union S i (i) of sets (i) with nite measure. And we suppose that there is a countable collection M o of measurable sets in which generate the collection of -measurable sets, in the sense that for every -measurable set X there is Y in the smallest -algebra containing M o so that X? X \ Y and Y? X \ Y are of measure zero. This is to avoid irrelevant trouble. To each s 2 assign a Hilbert space V s with norm j j s and inner product h; i s. This assignment is a Hilbert (space) bundle V over base space. The ber of the bundle V at s 2 is the Hilbert space V s. A section x of fv s : s 2 g is a function x :! a s2 V s so that x(s) 2 V s. We require also that there be chosen a collection F of sections so that For all x; y 2 F the C -valued function s! hx(s); y(s)i s is -measurable. If z is a section of fv s : s 2 g so that for all x 2 F the function s! hx(s); z(s)i is -measurable, then z 2 F. There is a countable collection of elements x 1 ; x 2 ; : : : in F so that for all s 2 the elements fx 1 (s); x 2 (s); : : :g are dense in V s.

3 Garrett: `A Good Spectral Theorem' 3 is a choice of measurable sections of V. Choice of F is part of the specication of a measurable Hilbert bundle. (There certainly may be many dierent choices of such collection of measurable sections). Note that separability is quite explicitly required. Sometimes fv s : s 2 g would be called a measurable eld of Hilbert spaces. We dene the norm of a measurable section x 2 F by jxj 2 = jx(s)j 2 s d(s) If this is zero, then x is a null section; if it is nite, then x is a square-integrable section. Let V s d(s) denote the collection of square-integrable sections modulo null sections. This is the direct integral of the Hilbert spaces V s, with respect to the measure, and with respect to (the choice) F of measurable sections. Proposition: The space R V s d(s) is a separable Hilbert space. Proof: This is very similar to the analogous proof for the `plain' L 2 (X; ) for an ordinary measure space (X; ). 3. Trivializing Hilbert bundles Here we show that direct integrals of Hilbert spaces are trivializable in a sense made precise just below. This fact is essential at certain technical points in the sequel, where we reduce various questions to the much easier analogues for trivial Hilbert bundles. We rst need to be explicit about morphisms (i.e., maps) of direct integrals of Hilbert spaces, or of Hilbert bundles, over a xed base space. Let fv s : s 2 g and : s 2 g be two such, with the same base space, with measurable sections F; F 0, respectively. Let ft s : s 2 g be a collection of continuous linear maps T s : V s! V 0 s. We evidently must assume that for v 2 F the section dened by fv 0 s (T v)(s) = T s (v(s)) is in F 0. Then we say that T preserves bers. If each T s is an isometry (or at least -almost-everywhere is such), then one easily checks that T is an isometry. If T s is -almost-everywhere an isomorphism of Hilbert spaces, then it is easy to show that T is such. And so on. Similarly, we can dene direct sums of direct integrals fv s g; fvsg 0 of Hilbert spaces over : the ber at s is just V s V 0 s, and the measurable sections are of the form s! v(s) v 0 (s) for v 2 cf and v 0 2 F 0. The norm on the direct sum is given by jv v 0 j 2 = jvj 2 + jv 0 j 2 where the latter two norms are those in the two original bundles. The norm also has an obvious integral expression.

4 4 Garrett: `A Good Spectral Theorem' The simplest of all direct integrals of Hilbert spaces is just L 2 (; ) itself, where V s = C with the usual inner product. Extending this example somewhat, a more general but still very simple type of direct integral is a direct integral of a constant or trivial Hilbert bundle, which by denition is one where V o = V s does not depend upon s 2. A direct integral of a constant Hilbert bundle may then be viewed as the space L 2 (; V o ; ) of square-integrable V o -valued functions on, which is dened to be the space of V o -valued functions f on so that jfj = hf(s); f(s)i Vo d(s) 1 2 < 1 A countable direct sum ^ i V i of Hilbert spaces V i (with inner products h; i i is dened to be the collection of sequences v = fv i g with v i 2 V i with (nite) norm dened by jvj X 2 = hv i ; v i i i F Generally, if is a countable disjoint union = i i of measurable subsets, then there is an obvious isometric isomorphism V s d(s) = ^ i i V s d(s) For simplicity of statement, we need a convention about cardinalities. In the remainder of this section we will refer to d-dimensional Hilbert spaces. Now, if d is a positive integer, this has the usual meaning, but if d = 1, we mean that the Hilbert space is separable but not nite-dimensional. Similarly, when we refer to a collection e ; : : : ; e d of vectors, if d is a positive integer this has the usual sense, while if d = 1 we mean this notation to refer to the countable collection e 1 ; e 2 ; : : : ; in which there is no actual element e 1. The following proposition eectively asserts that any direct integral of Hilbert spaces is isomorphic to a countable direct sum of direct integrals of trivial Hilbert bundles, where the sum is indexed only by dimension. An isomorphism of a direct integral to a direct integral of a trivial Hilbert bundle is called a trivialization of the bundle. Proposition: Let V = fv s : s 2 g be a Hilbert bundle with measurable sections F. There are measurable subsets d with indices d = 1; 2; : : : ; 1 so that for s 2 d the dimension of V s is d. For each d 1, there is a d-dimensional Hilbert space V (d) o so that we have an isometric isomorphism d V s d(s) L 2 ( d ; V (d) o ; ) which preserves bers and preserves inner products. Remarks: The proof contains a little more information than the statement, in that a semi-constructive `procedure' is described by which we obtain the desired isometric isomorphism. There are many possible dierent version of such a procedure, and it may be both more edifying and more comprehensible if the reader simply nds their own version, rather than read the version here.

5 Garrett: `A Good Spectral Theorem' 5 Proof: Let v 1 ; v 2 ; : : : be a countable collection of sections in F so that v 1 (s); v 2 (s); : : : is dense in the ber V s at s, for all s 2. For xed set I of indices, let I (s) be the absolute value of the determinant of the matrix of inner products hv i (s); v j (s)i s for i; j 2 I. (The choice of ordering of I doesn't matter since we take absolute value). This function is measurable. The vectors fv i (s) : i 2 Ig are linearly independent exactly where I (s) 6= 0. Let d be the supremum of functions I as I varies over all sts of indices with cardinality d. This function is also measurable. Then dim V s < d if and only if d (s) = 0. For 1 d < 1, let d be the measurable set of s where d (s) 6= 0 but d+1 (s) = 0. For d = 1, let d be the set of s where no d vanishes. These are the desired subsets as in the assertion of the proposition. It is easy to check that the restriction to each d of F provides a suitable class of measurable sections for d V s d(s) So now suppose that we have a Hilbert bundle all of whose bers are of the same dimension d 1. We will make a collection e 1 ; e 2 ; : : : ; e d of sections in F so that for all s 2 the collection e 1 (s); : : : ; e d (s) is an orthonormal basis for V s. This will give an isomorphism to a trivial bundle, as follows. Let C d be the standard d-dimensional Hilbert space, where by convention d = 1 gives the usual space `2 of square-integrable sequences. For a measurable section v, we have a map to C d -valued functions on given by v! (s! fhv(s); e i (s)i s : i = 1; 2; : : : ; dg) It is easy to check that if v is square-integrable then the associated `d-valued function is square integrable. That is, we have the desired isomorphism to L 2 (; C d ; ). Again, let v 1 ; v 2 ; : : : be the countable collection whose values pointwise everywhere are dense in the bers. We will `improve' v 1 so that jv 1 (s)j s = 1 for all s 2, basically by adding to it sections of the form fv i where f 2 L 1 (; ) and i > 1. (By refering to the denition one easily checks that such fv 2 F if f 2 L 1 (; ) and v 2 F). First, we normalize the v i. Let f i (s) = 1 for jv i (s)j = 0 and f i (s) = 1=jv i j for v i (s) 6= 0. Replacing v i by f i v i makes jv i (s)j s 1 for all s. Since we are only concerned with measurability and not square-integrability at the moment, we do not care about the nature of these functions f i beyond the fact that they are measurable. Now let i be the characteristic function of the set of s 2 where v 1 (s) = 0; : : : ; v i?1 (s) = 0 and put v 0 1 = v 1 + 2?1 2 v 2 + 2?2 3 v 3 + : : : + 2?n n v n + : : : By construction, the sequence of partial sums is convergent, the closure of the span of v 0 1 ; v 2; : : : is still the whole of V s at every point, and for all s 2 we have v 0 1(s) 6= 0. Indeed, for v 0 1(s) to vanish requires that all the v i (s) vanish, which is impossible. And then we may as well replace v 0 1 by e 1 (s) = v 0 1(s)=jv 0 1(s)j s to obtain a measurable section e 1 which is pointwise everywhere a unit vector.

6 6 Garrett: `A Good Spectral Theorem' Replace the sections v 2 ; v 3 ; ldots by the results obtained by applying Gram- Schmidt pointwise: replace v i by v(s)? hv i (s); e 1 (s)i s e 1 (s) Then we can repeat the previous process as applied to v 2, replacing it by a section e 2 pointwise everywhere a unit vector orthogonal to e 1. By induction, we use this procedure to construct a sequence e i of measurable sections so that for every s 2 the vectors e i (s) are an orthonormal basis for the ber V s, and in particular so that none of these sections vanish at any point s 2. Note that the procedure stops if at any point the Gram-Schmidt process causes all the subsequent sections to become zero. This will be the case if the bers are all of a xed nite dimension. Generally, many of the original sections v i will become identically zero along the way, but this does not harm the outcome. 4. Decomposable operators For each s 2, let T (s) be a bounded operator on the Hilbert space V s. If the function s! ht (s)x(s); y(s)i is a measurable C -valued function for all x; y 2 F, then say that ft s : s 2 g is a measurable eld of operators. If the integral converges, then we can form T s d(s) It turns out that we want to require convergence in the strong topology, and not in the uniform topology. Some especially simple measurable elds of operators are the multipliers: for f 2 L 1 (; ) let M f be the operator given simply by multiplication by f(s) on V s. Here of course we have chosen a function representing the equivalence class of f in L 1 (; ), but this does not aect the multiplier operator M f = obtained on the direct integral space. f(s) id Vs d(s) Theorem: Let M R be the collection of all multipliers M f on V = V s where f 2 L 1 (; ). Let D be the collection of decomposable operators on V. We have commutant relations M 0 = D D 0 = M Proof: We can invoke the results above which express a direct integral as a direct sum (indexed by dimension d of bers) of trivial direct integrals L 2 ( d ; V; ) = V d(s) since the characteristic functions of the subsets d where the bers are d-dimensional are measurable. d

7 Garrett: `A Good Spectral Theorem' 7 Note that the projectors to the subspaces where the bers have a xed dimension are given by multiplication by the characteristic functions of the subsets. That is, these projectors lie in M, and so commute with T 2 M. Thus, without loss of generality, we can restrict our attention to the situation that all bers have the same dimension. Then, by the previous section, we may suppose that the Hilbert bundle is trivial, consisting of the space L 2 (; V; ) of square-integrable V -valued functions on (modulo those functions almost everywhere zero, of course), where V is a (separable) Hilbert space. For v 2 V and for scalar-valued 2 L 2 (; ), we use the standard notation v for the function s! (s)v. Given an operator T 2 M 0, dene a eld ft s g of (ber-wise) operators by T s (v) = (T ( v))(s) for 2 L 1 (; ) having compact support and so that (s) = 1. Note that we make essential use of the triviality of the bundle in writing such an expression. We claim that the integral R Ts is none other than T itself. Proposition: The strong operator topology closure of the collection of multiplier operators M f with continuous f is the collection of multiplier operators M f with (measurable) essentially bounded f. Proof: Later commutative von Neumann Algebras Theorem: Let A be a commutative von Neumann algebra in B(V ) for a separable Hilbert space V. Let A be a C -algebra containing 1 and generated by countably-many elements and strong-operator topology dense in A. Let? : A! C o () be the Gelfand isomorphism. Then there is a positive regular Borel measure (described below) on so that? extends to an isomorphism ~? : A! L 1 (; ) There is x o 2 V so that A 0 x o is dense in V, where A 0 is the commutant Lemma: of A. That is, there is a cyclic vector x o for the commutant A 0. Corollary: The functional C o () 3 f! h??1 f(x o ); x o i on C o () is positive, so gives rise to an outer regular positive Borel measure on. Since V is separable, the conclusion of Banach-Alaoglu can be sharpened to see that is metrizeable, so is necessarily regular. Lemma: Any other choice of cyclic vector gives rise to a measure absolutely continuous with respect to. Thus, the denition of L 1 (; ) is independent of the choice of cyclic vector.

8 8 Garrett: `A Good Spectral Theorem' Theorem: Let A be a commutative von Neumann algebra in B(V ) for a separable Hilbert space V. Then there is a Hilbert bundle fv s : s 2 g over a compact metric space so that there is an isometry of Hilbert spaces and so that the map : V s d(s)! V L 1 (; ) 3 f! M f?1 is an isomorphism. Now we describe the Hilbert bundle which occurs in the theorem. The heuristic is that in the prototypical case where is the spectrum of a normal operator T, the ber V s is the s-`eigenspace' of T. Of course, this cannot be quite right, since there may be no eigenvalues whatsoever. For any x; y 2 V invoke the Riesz-Markov-Kakutani representation theorem to obtain a regular Borel measure x;y from the functional C o () 3 f! h??1 f(x); yi A crucial point is that x;y is absolutely continuous with respect to = xo;x o where x o is a cyclic vector for A 0. Thus, h x;y = d x;y d xo;x o 2 L 1 (; ) We have the crucial property that for f 2 C o () and x; y 2 V, letting T =??1 f 2 B(V ), h T f (x);y = f h x;y Let X be a countable dense subset of V, and let o be a subset of (diering from by a set of measure zero) so that for s 2 o and x; y 2 X the function values h x;y (s) make sense. Let V 0 be the C -linear span of??1 C o ()X inside V. Then h x;y (s) makes sense for s 2 o and x; y 2 V 0. Dene (x; y) s = h x;y (s) This is positive semi-denite on V 0. Let K s be the kernel of this hermitian form. Then (; ) s induces a positive-denite hermitian form denoted h; i s on the quotient V 0 =K s. Let V s be the Hilbert space obtained by completing this space with respect to the induced metric. For s 2 o and x 2 X, let ~x(s) be the image of x in V 0 =K s V s under the quotient map. Then fv s : s 2 o g is a Hilbert bundle over o with sections ~x for x 2 X. Dene F to be the collection of sections y so that is measurable for every x 2 X. Then s! h~x(s); y(s)i V o V s d(s) Since o diers from by a measure-zero set, the distinction is negligible.

9 Garrett: `A Good Spectral Theorem' 9 6. Spectral Theorem for a Normal Operator Let T be a normal operator on a separable Hilbert space V. Let A be the C - algebra generated by 1; T; T, i.e., the uniform operator topology closure of the polynomial ring C [T; T ]. Let A be the strong operator topology closure of A: this is a von Neumann algebra and is the double commutant A 00 of A and of C [T; T ], by the von Neumann density theorem. We may identify the maximal ideal space of A with the spectrum (T ) of T. Let? be the Gelfand isomorphism From above, we have an extension? : A! C o () ~? : A! L 1 (; ) where = xo;x o is the positive regular Borel measure attached to a cyclic vector x o for the commutant A 0. For any -measurable subset! of we have the associated e = e! = ~??1 ch! where ch! is the characteristic function of!. If! is of positive measure, then e = e! is a non-zero idempotent lying in the strong closure A 00 of C [T; T ]. The function dened by E : f measurable subsets of g! f self-adjoint projections on V g E(!) = ~??1 (ch! ) 2 B(V ) is easily obtained from ~??1. This function E is sometimes called a `resolution of the identity'. Note that the properties of the extended Gelfand transform ~? immediately yield the required properties of a `resolution of the identity': At the extremes, E(;) = 0 and E() = 1 V. Each E(!) is a self-adjoint projection. For mesaureable subsets!;! 0, we have E(! \! 0 ) = E(!) E(! 0 ) If! \! 0 = ; then E(! [! 0 ) = E(!) + E(! 0 ) For all x; y 2 V the function is a complex measure on. Indeed, for the last property, note that in our earlier notation. E x;y (!) := he(!)x; yi he(!)x; yi = x;y (!)

### 08a. Operators on Hilbert spaces. 1. Boundedness, continuity, operator norms

(February 24, 2017) 08a. Operators on Hilbert spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 2016-17/08a-ops

### 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

### 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

### Factorization of unitary representations of adele groups Paul Garrett garrett/

(February 19, 2005) Factorization of unitary representations of adele groups Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ The result sketched here is of fundamental importance in

### Elementary linear algebra

Chapter 1 Elementary linear algebra 1.1 Vector spaces Vector spaces owe their importance to the fact that so many models arising in the solutions of specific problems turn out to be vector spaces. The

### CHAPTER X THE SPECTRAL THEOREM OF GELFAND

CHAPTER X THE SPECTRAL THEOREM OF GELFAND DEFINITION A Banach algebra is a complex Banach space A on which there is defined an associative multiplication for which: (1) x (y + z) = x y + x z and (y + z)

### 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

### 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

### Analysis Preliminary Exam Workshop: Hilbert Spaces

Analysis Preliminary Exam Workshop: Hilbert Spaces 1. Hilbert spaces A Hilbert space H is a complete real or complex inner product space. Consider complex Hilbert spaces for definiteness. If (, ) : H H

### 1 Functional Analysis

1 Functional Analysis 1 1.1 Banach spaces Remark 1.1. In classical mechanics, the state of some physical system is characterized as a point x in phase space (generalized position and momentum coordinates).

### Math 350 Fall 2011 Notes about inner product spaces. In this notes we state and prove some important properties of inner product spaces.

Math 350 Fall 2011 Notes about inner product spaces In this notes we state and prove some important properties of inner product spaces. First, recall the dot product on R n : if x, y R n, say x = (x 1,...,

### CHAPTER VIII HILBERT SPACES

CHAPTER VIII HILBERT SPACES DEFINITION Let X and Y be two complex vector spaces. A map T : X Y is called a conjugate-linear transformation if it is a reallinear transformation from X into Y, and if T (λx)

### Linear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) 1.1 The Formal Denition of a Vector Space

Linear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) Contents 1 Vector Spaces 1 1.1 The Formal Denition of a Vector Space.................................. 1 1.2 Subspaces...................................................

### Lattice Theory Lecture 4. Non-distributive lattices

Lattice Theory Lecture 4 Non-distributive lattices John Harding New Mexico State University www.math.nmsu.edu/ JohnHarding.html jharding@nmsu.edu Toulouse, July 2017 Introduction Here we mostly consider

### 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

### 01. Review of metric spaces and point-set topology. 1. Euclidean spaces

(October 3, 017) 01. Review of metric spaces and point-set topology Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 017-18/01

### LECTURE VI: SELF-ADJOINT AND UNITARY OPERATORS MAT FALL 2006 PRINCETON UNIVERSITY

LECTURE VI: SELF-ADJOINT AND UNITARY OPERATORS MAT 204 - FALL 2006 PRINCETON UNIVERSITY ALFONSO SORRENTINO 1 Adjoint of a linear operator Note: In these notes, V will denote a n-dimensional euclidean vector

### Finite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product

Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )

### 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

### Garrett: `Bernstein's analytic continuation of complex powers' 2 Let f be a polynomial in x 1 ; : : : ; x n with real coecients. For complex s, let f

1 Bernstein's analytic continuation of complex powers c1995, Paul Garrett, garrettmath.umn.edu version January 27, 1998 Analytic continuation of distributions Statement of the theorems on analytic continuation

### SPECTRAL THEORY EVAN JENKINS

SPECTRAL THEORY EVAN JENKINS Abstract. These are notes from two lectures given in MATH 27200, Basic Functional Analysis, at the University of Chicago in March 2010. The proof of the spectral theorem for

### 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

### Math 413/513 Chapter 6 (from Friedberg, Insel, & Spence)

Math 413/513 Chapter 6 (from Friedberg, Insel, & Spence) David Glickenstein December 7, 2015 1 Inner product spaces In this chapter, we will only consider the elds R and C. De nition 1 Let V be a vector

### HILBERT SPACES AND THE RADON-NIKODYM THEOREM. where the bar in the first equation denotes complex conjugation. In either case, for any x V define

HILBERT SPACES AND THE RADON-NIKODYM THEOREM STEVEN P. LALLEY 1. DEFINITIONS Definition 1. A real inner product space is a real vector space V together with a symmetric, bilinear, positive-definite mapping,

### 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

### Part III. 10 Topological Space Basics. Topological Spaces

Part III 10 Topological Space Basics Topological Spaces Using the metric space results above as motivation we will axiomatize the notion of being an open set to more general settings. Definition 10.1.

### Stone-Čech compactification of Tychonoff spaces

The Stone-Čech compactification of Tychonoff spaces Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto June 27, 2014 1 Completely regular spaces and Tychonoff spaces A topological

### NOTES ON PRODUCT SYSTEMS

NOTES ON PRODUCT SYSTEMS WILLIAM ARVESON Abstract. We summarize the basic properties of continuous tensor product systems of Hilbert spaces and their role in non-commutative dynamics. These are lecture

### Vector Spaces. Vector space, ν, over the field of complex numbers, C, is a set of elements a, b,..., satisfying the following axioms.

Vector Spaces Vector space, ν, over the field of complex numbers, C, is a set of elements a, b,..., satisfying the following axioms. For each two vectors a, b ν there exists a summation procedure: a +

### Quadratic reciprocity (after Weil) 1. Standard set-up and Poisson summation

(December 19, 010 Quadratic reciprocity (after Weil Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ I show that over global fields k (characteristic not the quadratic norm residue symbol

### COMMON COMPLEMENTS OF TWO SUBSPACES OF A HILBERT SPACE

COMMON COMPLEMENTS OF TWO SUBSPACES OF A HILBERT SPACE MICHAEL LAUZON AND SERGEI TREIL Abstract. In this paper we find a necessary and sufficient condition for two closed subspaces, X and Y, of a Hilbert

### Lemma 1.3. The element [X, X] is nonzero.

Math 210C. The remarkable SU(2) Let G be a non-commutative connected compact Lie group, and assume that its rank (i.e., dimension of maximal tori) is 1; equivalently, G is a compact connected Lie group

### October 25, 2013 INNER PRODUCT SPACES

October 25, 2013 INNER PRODUCT SPACES RODICA D. COSTIN Contents 1. Inner product 2 1.1. Inner product 2 1.2. Inner product spaces 4 2. Orthogonal bases 5 2.1. Existence of an orthogonal basis 7 2.2. Orthogonal

### A VERY BRIEF REVIEW OF MEASURE THEORY

A VERY BRIEF REVIEW OF MEASURE THEORY A brief philosophical discussion. Measure theory, as much as any branch of mathematics, is an area where it is important to be acquainted with the basic notions and

### 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..............................

### Lifting to non-integral idempotents

Journal of Pure and Applied Algebra 162 (2001) 359 366 www.elsevier.com/locate/jpaa Lifting to non-integral idempotents Georey R. Robinson School of Mathematics and Statistics, University of Birmingham,

### Review of Linear Algebra

Review of Linear Algebra Throughout these notes, F denotes a field (often called the scalars in this context). 1 Definition of a vector space Definition 1.1. A F -vector space or simply a vector space

### Definitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations

Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of

### 5 Banach Algebras. 5.1 Invertibility and the Spectrum. Robert Oeckl FA NOTES 5 19/05/2010 1

Robert Oeckl FA NOTES 5 19/05/2010 1 5 Banach Algebras 5.1 Invertibility and the Spectrum Suppose X is a Banach space. Then we are often interested in (continuous) operators on this space, i.e, elements

### Representations of moderate growth Paul Garrett 1. Constructing norms on groups

(December 31, 2004) Representations of moderate growth Paul Garrett Representations of reductive real Lie groups on Banach spaces, and on the smooth vectors in Banach space representations,

### Linear Algebra 2 Spectral Notes

Linear Algebra 2 Spectral Notes In what follows, V is an inner product vector space over F, where F = R or C. We will use results seen so far; in particular that every linear operator T L(V ) has a complex

### Vector Space Basics. 1 Abstract Vector Spaces. 1. (commutativity of vector addition) u + v = v + u. 2. (associativity of vector addition)

Vector Space Basics (Remark: these notes are highly formal and may be a useful reference to some students however I am also posting Ray Heitmann's notes to Canvas for students interested in a direct computational

### Cover Page. The handle holds various files of this Leiden University dissertation.

Cover Page The handle http://hdl.handle.net/1887/22043 holds various files of this Leiden University dissertation. Author: Anni, Samuele Title: Images of Galois representations Issue Date: 2013-10-24 Chapter

### Quadratic reciprocity (after Weil) 1. Standard set-up and Poisson summation

(September 17, 010) Quadratic reciprocity (after Weil) Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ I show that over global fields (characteristic not ) the quadratic norm residue

### Chapter 8 Integral Operators

Chapter 8 Integral Operators In our development of metrics, norms, inner products, and operator theory in Chapters 1 7 we only tangentially considered topics that involved the use of Lebesgue measure,

### Review and problem list for Applied Math I

Review and problem list for Applied Math I (This is a first version of a serious review sheet; it may contain errors and it certainly omits a number of topic which were covered in the course. Let me know

### 1 Inner Product Space

Ch - Hilbert Space 1 4 Hilbert Space 1 Inner Product Space Let E be a complex vector space, a mapping (, ) : E E C is called an inner product on E if i) (x, x) 0 x E and (x, x) = 0 if and only if x = 0;

### Gaussian automorphisms whose ergodic self-joinings are Gaussian

F U N D A M E N T A MATHEMATICAE 164 (2000) Gaussian automorphisms whose ergodic self-joinings are Gaussian by M. L e m a ńc z y k (Toruń), F. P a r r e a u (Paris) and J.-P. T h o u v e n o t (Paris)

### Bounded and continuous functions on a locally compact Hausdorff space and dual spaces

Chapter 6 Bounded and continuous functions on a locally compact Hausdorff space and dual spaces Recall that the dual space of a normed linear space is a Banach space, and the dual space of L p is L q where

### C.6 Adjoints for Operators on Hilbert Spaces

C.6 Adjoints for Operators on Hilbert Spaces 317 Additional Problems C.11. Let E R be measurable. Given 1 p and a measurable weight function w: E (0, ), the weighted L p space L p s (R) consists of all

### 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

### Two-sided multiplications and phantom line bundles

Two-sided multiplications and phantom line bundles Ilja Gogić Department of Mathematics University of Zagreb 19th Geometrical Seminar Zlatibor, Serbia August 28 September 4, 2016 joint work with Richard

### OPERATOR THEORY ON HILBERT SPACE. Class notes. John Petrovic

OPERATOR THEORY ON HILBERT SPACE Class notes John Petrovic Contents Chapter 1. Hilbert space 1 1.1. Definition and Properties 1 1.2. Orthogonality 3 1.3. Subspaces 7 1.4. Weak topology 9 Chapter 2. Operators

### THE PROBLEMS FOR THE SECOND TEST FOR BRIEF SOLUTIONS

THE PROBLEMS FOR THE SECOND TEST FOR 18.102 BRIEF SOLUTIONS RICHARD MELROSE Question.1 Show that a subset of a separable Hilbert space is compact if and only if it is closed and bounded and has the property

### 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

### Recall that any inner product space V has an associated norm defined by

Hilbert Spaces Recall that any inner product space V has an associated norm defined by v = v v. Thus an inner product space can be viewed as a special kind of normed vector space. In particular every inner

### Math 676. A compactness theorem for the idele group. and by the product formula it lies in the kernel (A K )1 of the continuous idelic norm

Math 676. A compactness theorem for the idele group 1. Introduction Let K be a global field, so K is naturally a discrete subgroup of the idele group A K and by the product formula it lies in the kernel

### Measurable Choice Functions

(January 19, 2013) Measurable Choice Functions Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/fun/choice functions.pdf] This note

### CHAPTER 3. Hilbert spaces

CHAPTER 3 Hilbert spaces There are really three types of Hilbert spaces (over C). The finite dimensional ones, essentially just C n, for different integer values of n, with which you are pretty familiar,

### Five Mini-Courses on Analysis

Christopher Heil Five Mini-Courses on Analysis Metrics, Norms, Inner Products, and Topology Lebesgue Measure and Integral Operator Theory and Functional Analysis Borel and Radon Measures Topological Vector

### 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

### MATH 113 SPRING 2015

MATH 113 SPRING 2015 DIARY Effective syllabus I. Metric spaces - 6 Lectures and 2 problem sessions I.1. Definitions and examples I.2. Metric topology I.3. Complete spaces I.4. The Ascoli-Arzelà Theorem

### Projection-valued measures and spectral integrals

Projection-valued measures and spectral integrals Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto April 16, 2014 Abstract The purpose of these notes is to precisely define

### Spectral Measures, the Spectral Theorem, and Ergodic Theory

Spectral Measures, the Spectral Theorem, and Ergodic Theory Sam Ziegler The spectral theorem for unitary operators The presentation given here largely follows [4]. will refer to the unit circle throughout.

### 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

### Lecture Notes on Operator Algebras. John M. Erdman Portland State University. Version March 12, 2011

Lecture Notes on Operator Algebras John M. Erdman Portland State University Version March 12, 2011 c 2010 John M. Erdman E-mail address: erdman@pdx.edu Contents Chapter 1. LINEAR ALGEBRA AND THE SPECTRAL

### 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

### Your first day at work MATH 806 (Fall 2015)

Your first day at work MATH 806 (Fall 2015) 1. Let X be a set (with no particular algebraic structure). A function d : X X R is called a metric on X (and then X is called a metric space) when d satisfies

### 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,

### Lecture notes: Applied linear algebra Part 1. Version 2

Lecture notes: Applied linear algebra Part 1. Version 2 Michael Karow Berlin University of Technology karow@math.tu-berlin.de October 2, 2008 1 Notation, basic notions and facts 1.1 Subspaces, range and

### INTRODUCTION TO LIE ALGEBRAS. LECTURE 2.

INTRODUCTION TO LIE ALGEBRAS. LECTURE 2. 2. More examples. Ideals. Direct products. 2.1. More examples. 2.1.1. Let k = R, L = R 3. Define [x, y] = x y the cross-product. Recall that the latter is defined

### Linear Topological Spaces

Linear Topological Spaces by J. L. KELLEY ISAAC NAMIOKA AND W. F. DONOGHUE, JR. G. BALEY PRICE KENNETH R. LUCAS WENDY ROBERTSON B. J. PETTIS W. R. SCOTT EBBE THUE POULSEN KENNAN T. SMITH D. VAN NOSTRAND

### Inner product on B -algebras of operators on a free Banach space over the Levi-Civita field

Available online at wwwsciencedirectcom ScienceDirect Indagationes Mathematicae 26 (215) 191 25 wwwelseviercom/locate/indag Inner product on B -algebras of operators on a free Banach space over the Levi-Civita

### Theorem 5.3. Let E/F, E = F (u), be a simple field extension. Then u is algebraic if and only if E/F is finite. In this case, [E : F ] = deg f u.

5. Fields 5.1. Field extensions. Let F E be a subfield of the field E. We also describe this situation by saying that E is an extension field of F, and we write E/F to express this fact. If E/F is a field

### Kernel Method: Data Analysis with Positive Definite Kernels

Kernel Method: Data Analysis with Positive Definite Kernels 2. Positive Definite Kernel and Reproducing Kernel Hilbert Space Kenji Fukumizu The Institute of Statistical Mathematics. Graduate University

### NORMS ON SPACE OF MATRICES

NORMS ON SPACE OF MATRICES. Operator Norms on Space of linear maps Let A be an n n real matrix and x 0 be a vector in R n. We would like to use the Picard iteration method to solve for the following system

### Math 429/581 (Advanced) Group Theory. Summary of Definitions, Examples, and Theorems by Stefan Gille

Math 429/581 (Advanced) Group Theory Summary of Definitions, Examples, and Theorems by Stefan Gille 1 2 0. Group Operations 0.1. Definition. Let G be a group and X a set. A (left) operation of G on X is

### 1. Pseudo-Eisenstein series

(January 4, 202) Spectral Theory for SL 2 (Z)\SL 2 (R)/SO 2 (R) Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ Pseudo-Eisenstein series Fourier-Laplace-Mellin transforms Recollection

### Functional Analysis II held by Prof. Dr. Moritz Weber in summer 18

Functional Analysis II held by Prof. Dr. Moritz Weber in summer 18 General information on organisation Tutorials and admission for the final exam To take part in the final exam of this course, 50 % of

### Commutative Banach algebras 79

8. Commutative Banach algebras In this chapter, we analyze commutative Banach algebras in greater detail. So we always assume that xy = yx for all x, y A here. Definition 8.1. Let A be a (commutative)

### 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.

### SYLLABUS. 1 Linear maps and matrices

Dr. K. Bellová Mathematics 2 (10-PHY-BIPMA2) SYLLABUS 1 Linear maps and matrices Operations with linear maps. Prop 1.1.1: 1) sum, scalar multiple, composition of linear maps are linear maps; 2) L(U, V

### 4.4. Orthogonality. Note. This section is awesome! It is very geometric and shows that much of the geometry of R n holds in Hilbert spaces.

4.4. Orthogonality 1 4.4. Orthogonality Note. This section is awesome! It is very geometric and shows that much of the geometry of R n holds in Hilbert spaces. Definition. Elements x and y of a Hilbert

### Real Analysis: Part II. William G. Faris

Real Analysis: Part II William G. Faris June 3, 2004 ii Contents 1 Function spaces 1 1.1 Spaces of continuous functions................... 1 1.2 Pseudometrics and seminorms.................... 2 1.3 L

### Lebesgue Measure on R n

CHAPTER 2 Lebesgue Measure on R n Our goal is to construct a notion of the volume, or Lebesgue measure, of rather general subsets of R n that reduces to the usual volume of elementary geometrical 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.

### 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

### Normed and Banach Spaces

(August 30, 2005) Normed and Banach Spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ We have seen that many interesting spaces of functions have natural structures of Banach spaces:

### 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.

### Spanning and Independence Properties of Finite Frames

Chapter 1 Spanning and Independence Properties of Finite Frames Peter G. Casazza and Darrin Speegle Abstract The fundamental notion of frame theory is redundancy. It is this property which makes frames

### Hilbert Spaces. Hilbert space is a vector space with some extra structure. We start with formal (axiomatic) definition of a vector space.

Hilbert Spaces Hilbert space is a vector space with some extra structure. We start with formal (axiomatic) definition of a vector space. Vector Space. Vector space, ν, over the field of complex numbers,

### Functional Analysis Review

Outline 9.520: Statistical Learning Theory and Applications February 8, 2010 Outline 1 2 3 4 Vector Space Outline A vector space is a set V with binary operations +: V V V and : R V V such that for all

### MATH 8253 ALGEBRAIC GEOMETRY WEEK 12

MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f

### Functional Analysis I

Functional Analysis I Course Notes by Stefan Richter Transcribed and Annotated by Gregory Zitelli Polar Decomposition Definition. An operator W B(H) is called a partial isometry if W x = X for all x (ker

### 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......................................

### Lebesgue Measure on R n

8 CHAPTER 2 Lebesgue Measure on R n Our goal is to construct a notion of the volume, or Lebesgue measure, of rather general subsets of R n that reduces to the usual volume of elementary geometrical sets

### Elements of Positive Definite Kernel and Reproducing Kernel Hilbert Space

Elements of Positive Definite Kernel and Reproducing Kernel Hilbert Space Statistical Inference with Reproducing Kernel Hilbert Space Kenji Fukumizu Institute of Statistical Mathematics, ROIS Department

### Exercises to Applied Functional Analysis

Exercises to Applied Functional Analysis Exercises to Lecture 1 Here are some exercises about metric spaces. Some of the solutions can be found in my own additional lecture notes on Blackboard, as the