# INF-SUP CONDITION FOR OPERATOR EQUATIONS

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 INF-SUP CONDITION FOR OPERATOR EQUATIONS LONG CHEN We study the well-posedness of the operator equation (1) T u = f. where T is a linear and bounded operator between two linear vector spaces. We give equivalent conditions on the existence and uniqueness of the solution and apply to variational problems to obtain the so-called inf- condition (also known as Babuska condition). When the linear system is in the saddle point form, we derive another set of inf- conditions (known as Brezzi conditions). We shall skip the subscript of the norm for different spaces. It should be clear from the context. 1. PRELIMINARY FROM FUNCTION ANALYSIS In this section we recall some basic facts in functional analysis, notablely three theorems: Hahn-Banach theorem, Closed Range Theorem, and Open Mapping Theorem. For detailed explanation and sketch of proofs, we refer to Chapter: Minimal Functional Analysis for Computational Mathematicians Spaces. A complete normed space will be called a Banach space. A complete inner product space will be called a Hilbert space. Completeness means every Cauchy sequence will have a limit and the limit is in the space. Completeness is a nice property so that we can safely take the limit. Functional analysis is studying L (U, V ): the linear space consisting of all linear operators between two vector spaces U and V. When U and V are topological vector spaces (TVS), the subspace B(U, V ) L (U, V ) consists of all continuous linear operators. An operator T is bounded if T maps bounded sets into bounded sets. When U and V are normed spaces, for a bounded operator, there exists a constant M s.t. T u M u, u U. The smallest constant M is defined as the norm of T. For T B(U, V ), T = u U, u =1 T u. With such norm, the space B(U, V ) becomes a normed vector space. One can easily show that, for a linear operator, T is continuous iff T is bounded. Indeed by the translation invariance and the linearity of T, it suffices to prove such result at 0 for which the proof is straightforward by definition. An important and special example is V = R. The space L (U, R) is called the (algebraic) dual space of U and denoted by U. For an operator T L (U, V ), it induces an operator T L (V, U ) by T f, u = f, T u, where, is the duality pair, and T is called the transpose of T. The continuous linear functional of a normed space V will be denoted by V, i.e., V = B(V, R) and the continuous transpose T is defined as T. The algebraic dual space only uses the linear structure while the continuous dual space needs a topology (to define the continuity). Note that since R is a Banach space, V is a Banach Date: April 23,

2 2 LONG CHEN space no matter V is or not. In general, when the target space is complete, we can define the limit of a sequence of operators and show that the space of bounded operators is also complete. Namely if U is a normed linear space and V is a Banach space, then B(U, V ) is Banach. Question: Why are the dual space and the dual operator so important? (1) For an inner product space, the inner product structure is quite useful. For normed linear spaces, the duality pair, : V V R can play the (partial) role of the inner product. (2) Since U and V are Banach spaces, T is nicer than T. Many theorems are available for continuous linear operators between Banach spaces Hahn-Banach Theorem. A subspace S of a linear space V is a subset such that itself is a linear space with the addition and the scalar product defined for V. For a normed TVS, a closed subspace means the subspace is also closed under the norm topology, i.e., for every convergent sequence, the limit also lies in the subspace. Theorem 1.1 (Hahn-Banach Extension). Let V be a normed linear space and S V a subspace. For any f S = B(S, R) it can be extended to f V = B(V, R) with preservation of norms. For a continuous linear functional defined on a subspace, the natural extension by density can extend the domain of the operator to the closure of S. So we can take the closure of S and consider closed subspaces only. The following corollary says that we can find a functional to separate a point with a closed subspace. Corollary 1.2. Let V be a normed linear space and S V a closed subspace. Let v V but v / S. Then there exists a f V such that f(s) = 0 and f(v) = 1 and f = dist 1 (v, S). The corollary is obvious in an inner product space. We can use the vector f = v Proj S v which is orthogonal to S and scale f with the distance such that f(v) = 1. The extension of f is through the inner product. Thanks to the Hahn-Banach theorem, we can prove it without the inner product structure. Proof. Consider the subspace S v = span(s, v). For any u S v, u = u s + λv with u s S, λ R, we define f(u) = λ and use Hahn-Banach theorem to extend the domain of f to V. Then f(s) = 0 and f(v) = 1 and it is not hard to prove the norm of f = 1/d. Another corollary resembles the Reisz representation theorem. Corollary 1.3. Let V be a normed linear space. For any v V, there exists a f V such that f(v) = v 2 and f = v. Proof. For a Hilbert space, we simply chose f v = v and for a norm space, we can apply Corollary 1.2 to S = {0} and rescale the obtained functional. The norm structure in Hahn-Banach theorem is not necessary. It can be relaxed to a sub-linear functional and the preservation of norm can be relaxed to an inequality.

3 INF-SUP CONDITION FOR OPERATOR EQUATIONS Closed Range Theorem. For an operator T : U V, denoted by R(T ) the range of T and N(T ) the null space of T. For a matrix A m n treating as a linear operator from R n to R m, there are four fundamental subspaces R(A), N(A T ) R m, R(A T ), N(A) R n and the following relation (named the fundamental theorem of linear algebra by G. Strang [2]) holds (2) (3) R(A) N(A T ) = R m, R(A T ) N(A) = R n. We shall try to generalize (2)-(3) to operators T B(U, V ) between normed/inner product spaces. For T B(U, V ), the null space N(T ) := {u U, T u = 0} is a closed subspace. For a subset S in a Hilbert space H, the orthogonal complement S := {u H, (u, v) = 0, v S} is a closed subspace. For Banach spaces, we do not have the inner product structure but can use the duality pair, : V V R to define an orthogonal complement in the dual space which is called annihilator. More specifically, for a subset S in a normed space V, the annihilator S = {f V, f, v = 0, v S}. Similarly for a subset F V, we define F = {v V, f, v = 0, f F }. Similar to the orthogonal complement, annihilators are closed subspaces. For a subset (not necessarily a subspace) S V, S S if V is an inner product space or S (S ) if V is a normed space. The equality holds if and only if S is a closed subspace (which can be proved using Hahn-Banach theorem). The space S or (S ) is the smallest closed subspace containing S. The range R(T ) is not necessarily closed even T is continuous. As two closed subspaces, the relation N(T ) = R(T ) can be easily proved by definition. But the relation R(T ) = N(T ) may not hold since N(T ) is closed but R(T ) may not. It is easy to show R(T ) N(T ). The equality holds if and only R(T ) is closed. Theorem 1.4 (Closed Range Theorem). Let U and V be Banach spaces and let T B(U, V ). Then the following conditions are equivalent (1) R(T ) is closed in V. (2) R(T ) is closed in U. (3) R(T ) = N(T ) (4) R(T ) = N(T ). Closeness is a nice property. An operator is closed if its graph is closed in the product space. More precisely, let T : U V be a function and the graph of T is G(T ) = {(u, T u) : u U} U V. Then T is closed if its graph G is closed in U V in the product topology. The definition of closed operators only uses the topology of the product space. The operator is not necessarily linear or continuous. One can easily show a linear and continuous operator is closed. When T L (U, V ) and U, V are Banach spaces, these two properties are equivalent which is known as the closed graph theorem. The range of a closed linear operator between Banach spaces (and thus continuous) is not necessarily closed. Just compare their definitions: Graph is closed: if (u n, T u n ) (u, v), then v = T u. Range is closed: if T u n v, then there exists a u such that v = T u. The difference is: in the second line, we do not know if u n converges or not. But in the first line, we assume such limit exists.

4 4 LONG CHEN 1.4. Open Mapping Theorem. The stability of the equation can be ensured by the open mapping theorem. Theorem 1.5 (Open Mapping Theorem). For T B(U, V ) and both U and V are Banach spaces. If T is onto, then T is open. More explanation. Need Baire Category Theorem. 2. INF-SUP CONDITION: BABUŠKA THEORY The well-posedness of the operator equation T u = f consists of three questions: existence, uniqueness, and stability Operator Equations. For the uniqueness, a useful criterion to check is wether T is bounded below. Lemma 2.1. Let U and V be Banach spaces. For T B(U, V ), the range R(T ) is closed and T is injective if and only if T is bounded below, i.e., there exists a positive constant c such that (4) T u c u, for all u U. Proof. Sufficient. If T u = 0, inequality (4) implies u = 0, i.e., T is injective. Choosing a convergent sequence {T u k }, by (4), we know {u k } is also a Cauchy sequence and thus converges to some u U. The continuity of T shows that T u k converges to T u and thus R(T ) is closed. Necessary. When the range R(T ) is closed, as a closed subspace of a Banach space, it is also Banach. As T is injective, T 1 is well defined on R(T ). Apply Open Mapping Theorem to T : U R(T ), we conclude T 1 is continuous. Then u = T 1 (T u) T 1 T u which implies (4) with constant c = T 1 1. A trivial answer to the existence of the solution to (1) is: if f R(T ), then it is solvable. When is it solvable for all f V? The answer is V = R(T ), i.e., T is surjective. A characterization can be obtained using the dual of T. Lemma 2.2. Let U and V be Banach spaces and let T B(U, V ). Then T is surjective if and only if T is an injection and R(T ) is closed. Proof. Sufficient. By closed range theorem, R(T ) is also closed. Suppose R(T ) V, i.e., there exists a v V but v / R(T ). By Hahn-Banach theorem, there exists a f V such that f(r(t )) = 0 and f(v) = 1. Then T f U satisfies (5) T f, u = f, T u = 0, u U. So T f = 0 which implies f = 0 contradicts with the fact f(v) = 1. Necessary. When T is surjective, i.e., R(T ) = V is closed. By closed range theorem, so is R(T ). We then show if T f = 0, then f = 0. Indeed by (5), f, T u = 0. As R(T ) = V, this equivalent to f, v = 0 for all v V, i.e., f = 0. Combination of Lemma 2.1 and 2.2, we obtain a useful criteria for the operator T to be surjective. Corollary 2.3. Let U and V be Banach spaces and let T B(U, V ). Then T is surjective if and only if T is bounded below, i.e. T f c f for all f V.

5 INF-SUP CONDITION FOR OPERATOR EQUATIONS Abstract Variational Problems. Let a(, ) : U V R be a bilinear form on two Banach spaces U and V, i.e., it is linear to each variable. It will introduce two linear operators by A : U V, and A : V U Au, v = u, A v = a(u, v). We consider the operator equation: Given a f V, find u U such that (6) Au = f in V, or equivalently a(u, v) = f, v for all v V. To begin with, we have to assume both A and A are continuous which can be derived from the continuity of the bilinear form. (C) The bilinear form a(, ) is continuous in the sense that a(u, v) C u v, for all u U, v V. The minimal constant satisfies the above inequality will be denoted by a. With this condition, it is easy to check that A and A are bounded operators and A = A = a. The following conditions discuss the existence and the uniqueness. (E) inf a(u, v) v V u U u v = α E > 0. (U) inf a(u, v) u U v V u v = α U > 0. Theorem 2.4. Assume the bilinear form a(, ) is continuous, i.e., (C) holds, the problem (6) is well-posed if and only if (E) and (U) hold. Furthermore if (E) and (U) hold, then and thus for the solution to Au = f A 1 = (A ) 1 = α 1 U u 1 α f. = α 1 E = α 1, Proof. We can interpret (E) as A v α E v for all v V which is equivalent to A is surjective. Similarly (U) is Au α U u which is equivalent to A is injective. So A : U V is isomorphism and by open mapping theorem, A 1 is bounded and it is not hard to prove the norm is α 1 U. Prove for A is similar. Let us take the inf- condition (E) as an example to show how to verify it. It is easy to show (E) is equivalent to (7) for any v V, there exists u U, s.t. a(u, v) α u v. We shall present a slightly different characterization of (E). With this characterization, it is transformed to a construction of a suitable function.

6 6 LONG CHEN Theorem 2.5. The inf- condition (E) is equivalent to that for any v V, there exists u U, such that (8) a(u, v) C 1 v 2, and u C 2 v. Proof. Obviously (8) will imply (7) with α = C 1 /C 2. We now prove (E) implies (8). For any v V, by Corollary 1.3, there exists f V s.t. f(v) = v 2 and f = v. Since A is onto, we can find u s.t. Au = f and by open mapping theorem, we can find a u with u α 1 E f = α 1 E v and a(u, v) = Au, v = f(v) = v 2. For a given v, the desired u satisfying (8) could dependent on v in a subtle way. A special and simple case is u = v when U = V which is known the coercivity. The corresponding result is known as Lax-Milgram Theorem. Corollary 2.6 (Lax-Milgram). For a bilinear form a(, ) on V V, if it satisfies (1) Continuity: a(u, v) β u v ; (2) Coercivity : a(u, u) α u 2, then for any f V, there exists a unique u V such that and a(u, v) = f, v, u β/α f. The simplest case is the bilinear form a(, ) is symmetric and positive definite on V. Then a(, ) defines a new inner product. Lax-Milgram theorem is simply the Riesz representation theorem Conforming Discretization of Variational Problems. We consider conforming discretizations of the variational problem (9) a(u, v) = f, v in the finite dimensional subspaces U h U and V h V. Find u h U h such that (10) a(u h, v h ) = f, v h, for all v h V h. The existence and uniqueness of (10) is equivalent to the following discrete inf- conditions: a(u h, v h ) (D) inf u U h v V h u h v h = inf a(u h, v h ) v V h u U h u h v h = α h > 0. With appropriate choice of basis, (10) has a matrix form. To be well defined, first of all the matrix should be square. Second the matrix should be full rank (non singular). For a squared matrix, two inf- conditions are merged into one. To be uniformly stable, the constant α h should be uniformly bounded below. An abstract error analysis can be established using inf- conditions. The key property for the conforming discretization is the following Garlerkin orthogonality a(u u h, v h ) = 0, for all v h V h. Theorem 2.7. If the bilinear form a(, ) satisfies (C), (E), (U) and (D), then there exists a unique solution u U to (9) and a unique solution u h U h to (10). Furthermore u u h a α h inf u v h. v h U h

7 INF-SUP CONDITION FOR OPERATOR EQUATIONS 7 Proof. With those assumptions, we know for a given f V, the corresponding solutions u and u h are well defined. Let us define a projection operator P h : U U h by P h u = u h. Note that P h Uh is identity. In operator form P h = A 1 h Q ha, where Q h : V V h is the natural inclusion of dual spaces. We prove that P h is a bounded linear operator and P h a /α h as the following: u h 1 a(u h, v h ) α h v h V h v h = 1 a(u, v h ) α h v h V h v h 1 a(u, v) α h v V v a u. α h Then for any w h U h, note that P h w h = w h, u u h = (I P )(u w h ) I P h u w h. Since P 2 h = P h, we use the identity in [3]: I P h = P h, to get the desired result. 3. INF-SUP CONDITIONS FOR SADDLE POINT SYSTEM: BREZZI THEORY 3.1. Variational problem in the mixed form. We shall consider an abstract mixed variational problem first. Let V and P be two Banach spaces. For given (f, g) V P, find (u, p) V P such that: (11) (12) a(u, v) + b(v, p) = f, v, for all v V, b(u, q) = g, q, for all q P. Let us introduce linear operators A : V V, as Au, v = a(u, v) and B : V P, B : P V, as Bv, q = v, B q = b(v, q). Written in the operator form, the problem becomes (13) (14) or in short (15) Au + B p = f, B u = g, ( ) ( ) A B u = B 0 p ( f g).

8 8 LONG CHEN 3.2. inf- conditions. We shall study the well posedness of this abstract mixed problem. First we assume all bilinear forms are continuous so that all operators A, B, B are continuous. (C) The bilinear form a(, ), and b(, ) are continuous a(u, v) C u v, for all u, v V, b(v, q) C v q, for all v V, q P. The solvable of the second equation (14) is equivalent to B is surjective or B is injective and R(B ) closed which is equivalent to the following inf- condition (B) inf b(v, q) q P v V v q = β > 0 With condition (B), we have B : V/N(B) P is an isomorphism. So given g P, we can chose u 1 V/N(B) such that Bu 1 = g and u 1 V β 1 g P. After we get a unique u 1, we restrict the test function v in (11) to N(B). Since v, B q = Bv, q = 0 for v N(B), we get the following variational form: find u 0 N(B) such that (16) a(u 0, v) = f, v a(u 1, v), for all v N(B). The existence and uniqueness of u 0 is then equivalent to the two inf- conditions for a(u, v) on space Z = N(B). (A) inf a(u, v) u Z v Z u v = inf a(u, v) v Z u Z u v = α > 0. After we determine a unique u = u 0 + u 1 in this way, we solve (17) B p = f Au to get p. Since u 0 is the solution to (16), the right hand side f Au N(B). Thus we require B : V N(B) is an isomorphism which is also equivalent to the condition (B). Theorem 3.1. Assume the bilinear forms a(, ), b(, ) are continuous, i.e., (C) holds. The mixed variational problem (15) is well-posed if and only if (A) and (B) hold. When (A) and (B) hold, we have the stability result u V + p P f V + g P. The following characterization of the inf- condition for the operator B is useful. The verification is again transferred to a construction of a suitable function. The proof is similar to that in Theorem 2.5 and thus skipped here. Theorem 3.2. The inf- condition (B) is equivalent to that: for any q P, there exists v V, such that (18) b(v, q) C 1 q 2, and v C 2 q. Note that in general a construction of desirable v = v(q), especially the control of norm v, may not be straightforward.

9 INF-SUP CONDITION FOR OPERATOR EQUATIONS Conforming Discretization. We consider finite element approximation to the mixed problem: Find u h V h and p h P h such that (19) (20) a(u h, v h ) + b(v h, p h ) = f, v h, for all v h V h, b(u h, q h ) = g, q h, for all q h P h. We shall mainly consider the conforming case V h V and P h P. We denote B h : V h P h which can be written as Q hbi h with natural embedding I h : V h V and Q h : P P h, and denote Z h = N(B h ). Recall that Z = N(B). In the application to Stokes equations B = div, so Z is called divergence free space and Z h is discrete divergence free space. Remark 3.3. In general Z h Z. Namely a discrete divergence free function may not be exactly divergence free. Just compare the meaning of B h u h = 0 in (P h ) B h u h, q h = 0, for all q h P h, with Bu h = 0 in P Bu h, q = 0, for all q P. If we can identify P = P and P h = (P h ) using Riesz representation theorem, then N(B h ) (P h ) which may contains non-trivial elements in P. Namely it is possible that Bu h ker(q h ) B(V h ). To enforce Z h Z, it suffices to have B(V h ) P h. Indeed when B(V h ) P h, Q h Bu h = Bu h and thus B h u h = 0 implies Bu h = 0. The discrete inf- conditions for the finite element approximation will be (D) (A h ) inf u h Z h (B h ) inf q h P h a(u h, v h ) = α h > 0, v h Z h u h V v h V b(v h, q h ) = β h > 0. v h V h v h V q h P Theorem 3.4. If (A), (B), (C) and (D) hold, then the discrete problem is well-posed and u u h V + p p h P C inf u v h V + p q h P. v h V h,q h P h Exercise 3.5. Let U = V P and rewrite the mixed formulation using one bilinear form defined on U. Then use Babuska theory to prove the above theorem. Write explicitly how the constant C depends on the constants in all inf- conditions Fortin operator. Note that the inf- condition (B) in the continuous level implies: for any q h P h, there exists v V such that b(v, q h ) β v V q h P and v C q h. For the discrete inf- condition, we need a v h V h satisfying such property. One approach is to use the so-called Fortin operator [1] to get such a v h from v. Definition 3.6 (Fortin operator). A linear operator Π h : V V h is called a Fortin operator if (1) b(π h v, q h ) = b(v, q h ) for all q h P h (2) Π h v V C v V. Theorem 3.7. Assume the inf- condition (B) holds and there exists a Fortin operator Π h, then the discrete inf- condition (B h ) holds.

10 10 LONG CHEN Proof. The inf- condition (B) in the continuous level implies: for any q h P h, there exists v V such that b(v, q h ) β v q h and v C q h. We choose v h = Π h v. By the definition of Fortin operator b(v h, q h ) = b(v, q h ) β v V q h P βc v h V q h P. The discrete inf- condition then follows. REFERENCES [1] M. Fortin. An analysis of the convergence of mixed finite element methods. RAIRO Anal. Numer, 11(R3): , [2] G. Strang. The Fundamental Theorem of Linear Algebra. American Mathematical Monthly, 100(9): , [3] J. Xu and L. Zikatanov. Some Observations on {Babu{š}ka} and {Brezzi} Theories. Numer. Math., 94(1): , mar 2003.

### 2.3 Variational form of boundary value problems

2.3. VARIATIONAL FORM OF BOUNDARY VALUE PROBLEMS 21 2.3 Variational form of boundary value problems Let X be a separable Hilbert space with an inner product (, ) and norm. We identify X with its dual X.

### Numerische Mathematik

Numer. Math. (2003) 94: 195 202 Digital Object Identifier (DOI) 10.1007/s002110100308 Numerische Mathematik Some observations on Babuška and Brezzi theories Jinchao Xu, Ludmil Zikatanov Department of Mathematics,

### Chapter 2 Finite Element Spaces for Linear Saddle Point Problems

Chapter 2 Finite Element Spaces for Linear Saddle Point Problems Remark 2.1. Motivation. This chapter deals with the first difficulty inherent to the incompressible Navier Stokes equations, see Remark

### 1 Constrained Optimization

1 Constrained Optimization Let B be an M N matrix, a linear operator from the space IR N to IR M with adjoint B T. For (column vectors) x IR N and y IR M we have x B T y = Bx y. This vanishes for all y

### 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

### Weak Formulation of Elliptic BVP s

Weak Formulation of Elliptic BVP s There are a large number of problems of physical interest that can be formulated in the abstract setting in which the Lax-Milgram lemma is applied to an equation expressed

### Approximation in Banach Spaces by Galerkin Methods

2 Approximation in Banach Spaces by Galerkin Methods In this chapter, we consider an abstract linear problem which serves as a generic model for engineering applications. Our first goal is to specify the

### WELL POSEDNESS OF PROBLEMS I

Finite Element Method 85 WELL POSEDNESS OF PROBLEMS I Consider the following generic problem Lu = f, where L : X Y, u X, f Y and X, Y are two Banach spaces We say that the above problem is well-posed (according

### 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

### Introduction to Empirical Processes and Semiparametric Inference Lecture 22: Preliminaries for Semiparametric Inference

Introduction to Empirical Processes and Semiparametric Inference Lecture 22: Preliminaries for Semiparametric Inference Michael R. Kosorok, Ph.D. Professor and Chair of Biostatistics Professor of Statistics

### 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

### Chapter 1 Foundations of Elliptic Boundary Value Problems 1.1 Euler equations of variational problems

Chapter 1 Foundations of Elliptic Boundary Value Problems 1.1 Euler equations of variational problems Elliptic boundary value problems often occur as the Euler equations of variational problems the latter

### 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

### 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

### 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

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

### 1 Definition and Basic Properties of Compa Operator

1 Definition and Basic Properties of Compa Operator 1.1 Let X be a infinite dimensional Banach space. Show that if A C(X ), A does not have bounded inverse. Proof. Denote the unit ball of X by B and the

### SPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS

SPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS TSOGTGEREL GANTUMUR Abstract. After establishing discrete spectra for a large class of elliptic operators, we present some fundamental spectral properties

### 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

### 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

### 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

### STOKES PROBLEM WITH SEVERAL TYPES OF BOUNDARY CONDITIONS IN AN EXTERIOR DOMAIN

Electronic Journal of Differential Equations, Vol. 2013 2013, No. 196, pp. 1 28. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu STOKES PROBLEM

### 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

### Introduction to Functional Analysis With Applications

Introduction to Functional Analysis With Applications A.H. Siddiqi Khalil Ahmad P. Manchanda Tunbridge Wells, UK Anamaya Publishers New Delhi Contents Preface vii List of Symbols.: ' - ix 1. Normed and

### Chapter 1. Preliminaries. The purpose of this chapter is to provide some basic background information. Linear Space. Hilbert Space.

Chapter 1 Preliminaries The purpose of this chapter is to provide some basic background information. Linear Space Hilbert Space Basic Principles 1 2 Preliminaries Linear Space The notion of linear space

### 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

### 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

### 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

### On Riesz-Fischer sequences and lower frame bounds

On Riesz-Fischer sequences and lower frame bounds P. Casazza, O. Christensen, S. Li, A. Lindner Abstract We investigate the consequences of the lower frame condition and the lower Riesz basis condition

### Spectral Properties of Elliptic Operators In previous work we have replaced the strong version of an elliptic boundary value problem

Spectral Properties of Elliptic Operators In previous work we have replaced the strong version of an elliptic boundary value problem L u x f x BC u x g x with the weak problem find u V such that B u,v

### The Dual of a Hilbert Space

The Dual of a Hilbert Space Let V denote a real Hilbert space with inner product, u,v V. Let V denote the space of bounded linear functionals on V, equipped with the norm, F V sup F v : v V 1. Then for

### 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

### 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,

### Reductions of Operator Pencils

Reductions of Operator Pencils Olivier Verdier Department of Mathematical Sciences, NTNU, 7491 Trondheim, Norway arxiv:1208.3154v2 [math.na] 23 Feb 2014 2018-10-30 We study problems associated with an

### 4 Linear operators and linear functionals

4 Linear operators and linear functionals The next section is devoted to studying linear operators between normed spaces. Definition 4.1. Let V and W be normed spaces over a field F. We say that T : V

### A Simple Proof of the Fredholm Alternative and a Characterization of the Fredholm Operators

thus a n+1 = (2n + 1)a n /2(n + 1). We know that a 0 = π, and the remaining part follows by induction. Thus g(x, y) dx dy = 1 2 tanh 2n v cosh v dv Equations (4) and (5) give the desired result. Remarks.

### 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

### NONSTANDARD NONCONFORMING APPROXIMATION OF THE STOKES PROBLEM, I: PERIODIC BOUNDARY CONDITIONS

NONSTANDARD NONCONFORMING APPROXIMATION OF THE STOKES PROBLEM, I: PERIODIC BOUNDARY CONDITIONS J.-L. GUERMOND 1, Abstract. This paper analyzes a nonstandard form of the Stokes problem where the mass conservation

### t y n (s) ds. t y(s) ds, x(t) = x(0) +

1 Appendix Definition (Closed Linear Operator) (1) The graph G(T ) of a linear operator T on the domain D(T ) X into Y is the set (x, T x) : x D(T )} in the product space X Y. Then T is closed if its graph

### Where is matrix multiplication locally open?

Linear Algebra and its Applications 517 (2017) 167 176 Contents lists available at ScienceDirect Linear Algebra and its Applications www.elsevier.com/locate/laa Where is matrix multiplication locally open?

### A Concise Course on Stochastic Partial Differential Equations

A Concise Course on Stochastic Partial Differential Equations Michael Röckner Reference: C. Prevot, M. Röckner: Springer LN in Math. 1905, Berlin (2007) And see the references therein for the original

### 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

### Inequalities of Babuška-Aziz and Friedrichs-Velte for differential forms

Inequalities of Babuška-Aziz and Friedrichs-Velte for differential forms Martin Costabel Abstract. For sufficiently smooth bounded plane domains, the equivalence between the inequalities of Babuška Aziz

### Functional Analysis HW #1

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

### Non commutative Khintchine inequalities and Grothendieck s theo

Non commutative Khintchine inequalities and Grothendieck s theorem Nankai, 2007 Plan Non-commutative Khintchine inequalities 1 Non-commutative Khintchine inequalities 2 µ = Uniform probability on the set

### Reflexivity of Locally Convex Spaces over Local Fields

Reflexivity of Locally Convex Spaces over Local Fields Tomoki Mihara University of Tokyo & Keio University 1 0 Introduction For any Hilbert space H, the Hermit inner product induces an anti C- linear isometric

### Chapter 3: Baire category and open mapping theorems

MA3421 2016 17 Chapter 3: Baire category and open mapping theorems A number of the major results rely on completeness via the Baire category theorem. 3.1 The Baire category theorem 3.1.1 Definition. A

### 1 The Stokes System. ρ + (ρv) = ρ g(x), and the conservation of momentum has the form. ρ v (λ 1 + µ 1 ) ( v) µ 1 v + p = ρ f(x) in Ω.

1 The Stokes System The motion of a (possibly compressible) homogeneous fluid is described by its density ρ(x, t), pressure p(x, t) and velocity v(x, t). Assume that the fluid is barotropic, i.e., the

### Contents. Preface for the Instructor. Preface for the Student. xvii. Acknowledgments. 1 Vector Spaces 1 1.A R n and C n 2

Contents Preface for the Instructor xi Preface for the Student xv Acknowledgments xvii 1 Vector Spaces 1 1.A R n and C n 2 Complex Numbers 2 Lists 5 F n 6 Digression on Fields 10 Exercises 1.A 11 1.B Definition

### 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

### 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

### 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

### Normed & Inner Product Vector Spaces

Normed & Inner Product Vector Spaces ECE 174 Introduction to Linear & Nonlinear Optimization Ken Kreutz-Delgado ECE Department, UC San Diego Ken Kreutz-Delgado (UC San Diego) ECE 174 Fall 2016 1 / 27 Normed

### Equations paraboliques: comportement qualitatif

Université de Metz Master 2 Recherche de Mathématiques 2ème semestre Equations paraboliques: comportement qualitatif par Ralph Chill Laboratoire de Mathématiques et Applications de Metz Année 25/6 1 Contents

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

### A NOTE ON THE LADYŽENSKAJA-BABUŠKA-BREZZI CONDITION

A NOTE ON THE LADYŽENSKAJA-BABUŠKA-BREZZI CONDITION JOHNNY GUZMÁN, ABNER J. SALGADO, AND FRANCISCO-JAVIER SAYAS Abstract. The analysis of finite-element-like Galerkin discretization techniques for the

### Intertibility and spectrum of the multiplication operator on the space of square-summable sequences

Intertibility and spectrum of the multiplication operator on the space of square-summable sequences Objectives Establish an invertibility criterion and calculate the spectrum of the multiplication operator

### Finite Elements. Colin Cotter. February 22, Colin Cotter FEM

Finite Elements February 22, 2019 In the previous sections, we introduced the concept of finite element spaces, which contain certain functions defined on a domain. Finite element spaces are examples of

### 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

### Contents. Index... 15

Contents Filter Bases and Nets................................................................................ 5 Filter Bases and Ultrafilters: A Brief Overview.........................................................

### 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

### LECTURE 1: SOURCES OF ERRORS MATHEMATICAL TOOLS A PRIORI ERROR ESTIMATES. Sergey Korotov,

LECTURE 1: SOURCES OF ERRORS MATHEMATICAL TOOLS A PRIORI ERROR ESTIMATES Sergey Korotov, Institute of Mathematics Helsinki University of Technology, Finland Academy of Finland 1 Main Problem in Mathematical

### 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

### Functional Analysis F3/F4/NVP (2005) Homework assignment 2

Functional Analysis F3/F4/NVP (25) Homework assignment 2 All students should solve the following problems:. Section 2.7: Problem 8. 2. Let x (t) = t 2 e t/2, x 2 (t) = te t/2 and x 3 (t) = e t/2. Orthonormalize

### 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

### Variational Formulations

Chapter 2 Variational Formulations In this chapter we will derive a variational (or weak) formulation of the elliptic boundary value problem (1.4). We will discuss all fundamental theoretical results that

### 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

### USING FUNCTIONAL ANALYSIS AND SOBOLEV SPACES TO SOLVE POISSON S EQUATION

USING FUNCTIONAL ANALYSIS AND SOBOLEV SPACES TO SOLVE POISSON S EQUATION YI WANG Abstract. We study Banach and Hilbert spaces with an eye towards defining weak solutions to elliptic PDE. Using Lax-Milgram

### 31. Successive Subspace Correction method for Singular System of Equations

Fourteenth International Conference on Domain Decomposition Methods Editors: Ismael Herrera, David E. Keyes, Olof B. Widlund, Robert Yates c 2003 DDM.org 31. Successive Subspace Correction method for Singular

### B. Appendix B. Topological vector spaces

B.1 B. Appendix B. Topological vector spaces B.1. Fréchet spaces. In this appendix we go through the definition of Fréchet spaces and their inductive limits, such as they are used for definitions of function

### A Brief Introduction to Functional Analysis

A Brief Introduction to Functional Analysis Sungwook Lee Department of Mathematics University of Southern Mississippi sunglee@usm.edu July 5, 2007 Definition 1. An algebra A is a vector space over C with

### Exercises on chapter 0

Exercises on chapter 0 1. A partially ordered set (poset) is a set X together with a relation such that (a) x x for all x X; (b) x y and y x implies that x = y for all x, y X; (c) x y and y z implies that

### Functional Analysis, Math 7321 Lecture Notes from April 04, 2017 taken by Chandi Bhandari

Functional Analysis, Math 7321 Lecture Notes from April 0, 2017 taken by Chandi Bhandari Last time:we have completed direct sum decomposition with generalized eigen space. 2. Theorem. Let X be separable

### Fredholm Theory. April 25, 2018

Fredholm Theory April 25, 208 Roughly speaking, Fredholm theory consists of the study of operators of the form I + A where A is compact. From this point on, we will also refer to I + A as Fredholm operators.

### 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

### 1. Subspaces A subset M of Hilbert space H is a subspace of it is closed under the operation of forming linear combinations;i.e.,

Abstract Hilbert Space Results We have learned a little about the Hilbert spaces L U and and we have at least defined H 1 U and the scale of Hilbert spaces H p U. Now we are going to develop additional

### 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.

### Multigrid Methods for Saddle Point Problems

Multigrid Methods for Saddle Point Problems Susanne C. Brenner Department of Mathematics and Center for Computation & Technology Louisiana State University Advances in Mathematics of Finite Elements (In

### Construction of some Generalized Inverses of Operators between Banach Spaces and their Selections, Perturbations and Applications

Ph. D. Dissertation Construction of some Generalized Inverses of Operators between Banach Spaces and their Selections, Perturbations and Applications by Haifeng Ma Presented to the Faculty of Mathematics

### Université de Metz. Master 2 Recherche de Mathématiques 2ème semestre. par Ralph Chill Laboratoire de Mathématiques et Applications de Metz

Université de Metz Master 2 Recherche de Mathématiques 2ème semestre Systèmes gradients par Ralph Chill Laboratoire de Mathématiques et Applications de Metz Année 26/7 1 Contents Chapter 1. Introduction

### Fall, 2003 CIS 610. Advanced geometric methods. Homework 3. November 11, 2003; Due November 25, beginning of class

Fall, 2003 CIS 610 Advanced geometric methods Homework 3 November 11, 2003; Due November 25, beginning of class You may work in groups of 2 or 3 Please, write up your solutions as clearly and concisely

ECE 275AB Lecture 5 Fall 2008 V1.1 c K. Kreutz-Delgado, UC San Diego p. 1/1 Lecture 5 ECE 275A Pseudoinverse and Adjoint Operators ECE 275AB Lecture 5 Fall 2008 V1.1 c K. Kreutz-Delgado, UC San Diego p.

### Notes on Integrable Functions and the Riesz Representation Theorem Math 8445, Winter 06, Professor J. Segert. f(x) = f + (x) + f (x).

References: Notes on Integrable Functions and the Riesz Representation Theorem Math 8445, Winter 06, Professor J. Segert Evans, Partial Differential Equations, Appendix 3 Reed and Simon, Functional 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

### 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

### A PROPERTY OF STRICTLY SINGULAR 1-1 OPERATORS

A PROPERTY OF STRICTLY SINGULAR - OPERATORS GEORGE ANDROULAKIS, PER ENFLO Abstract We prove that if T is a strictly singular - operator defined on an infinite dimensional Banach space X, then for every

### 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

### 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

### Fact Sheet Functional Analysis

Fact Sheet Functional Analysis Literature: Hackbusch, W.: Theorie und Numerik elliptischer Differentialgleichungen. Teubner, 986. Knabner, P., Angermann, L.: Numerik partieller Differentialgleichungen.

### Statistics 612: L p spaces, metrics on spaces of probabilites, and connections to estimation

Statistics 62: L p spaces, metrics on spaces of probabilites, and connections to estimation Moulinath Banerjee December 6, 2006 L p spaces and Hilbert spaces We first formally define L p spaces. Consider

### David Hilbert was old and partly deaf in the nineteen thirties. Yet being a diligent

Chapter 5 ddddd dddddd dddddddd ddddddd dddddddd ddddddd Hilbert Space The Euclidean norm is special among all norms defined in R n for being induced by the Euclidean inner product (the dot product). A

### Traces and Duality Lemma

Traces and Duality Lemma Recall the duality lemma with H / ( ) := γ 0 (H ()) defined as the trace space of H () endowed with minimal extension norm; i.e., for w H / ( ) L ( ), w H / ( ) = min{ ŵ H () ŵ

### 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,...,

### 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

### 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 (, )

### Chapter 2 Linear Transformations

Chapter 2 Linear Transformations Linear Transformations Loosely speaking, a linear transformation is a function from one vector space to another that preserves the vector space operations. Let us be more

### 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

### 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