DOCSLIB.ORG

Functional Analysis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Functional Analysis Functional Analysis Irena Swanson, Reed College, Spring 2011 This is work in progress. I am still adding, subtracting, modifying. Any comments, solutions, corrections are welcome. Table of contents: Section 1: Overview, background 1 Section 2: Definition of Banach spaces 2 Section 3: Examples of Banach spaces 2 Section 4: Lp spaces 6 Section 5: L∞ spaces 9 Section 6: Some spaces that are almost Banach, but aren’t 10 Section 7: Normed vector spaces are metric spaces 11 Section 8: How to make new spaces out of existent ones (or not) 13 Section 9: Finite-dimensional vector spaces are special 18 Section 10: Examples of (infinite dimensional) subspaces 20 Section 11: Sublinear functional and the Hahn–Banach Theorem 22 Section 12: A section of big theorems 27 Section 13: Hilbert spaces 35 Section 14: Perpendicularity 37 Section 15: The Riesz Representation Theorem 41 Section 16: Orthonormal bases 41 Section 17: Section I.5 in the book 46 Section 18: Adjoints of linear transformations 47 Section 19: Self-adjoint operators 49 Section 20: Spectrum 51 Section 21: Holomorphic Banach space-valued functions 55 Appendix A: Some topology facts 59 Appendix B: Homework and exam sets 60 1 1 Overview, background Throughout, F will stand for either R or C. Thus F is a complete (normed) field. [Conway’s book assumes that all topological spaces are Hausdorff. Let’s try to not impose that unless needed.] Much of the material and inspiration came from Larry Brown’s lectures on functional analysis at Purdue University in the 1990s, and some came from my Reed thesis 1987. Functional analysis is a wonderful blend of analysis and algebra, of finite-dimensional and infinite-dimensional, so it is interesting, versatile, useful. I will cover Banach spaces first, Hilbert spaces second, as Banach spaces are more general. 2 Definition of Banach spaces Definition 2.1 Let X be a vector space over F. A norm on X is a function : X || || → [0, ) such that ∞ (1) (Positive semidefiniteness) For all x X, x =0 if and only if x =0. ∈ || || (2) rx = r x for all x X and all r F. || || | | || || ∈ ∈ (3) (Triangle inequality) For all x,y X, x + y x + y . ∈ || || ≤ || || || || Is is easy to verify that for all x,y X, x y x y , and that x y ∈ || − || ≤ || || − || || | || || − || ||| ≤ x y . || ± || Note that if X is a normed vector space, then X has a metric: d(x,y) = x y . || − || Definition 2.2 A vector space X with a norm is called a normed vector space or || || a normed linear space. In a normed linear space, a sequence f is Cauchy (in the { n} norm) if for all ǫ > 0 there exists N such that for all m, n N, f f < ǫ. A sequence ≥ || n − m|| f converges (in the norm) to f if for all ǫ > 0 there exists N such that for all n N, { n} ≥ f f < ǫ. A sequence f in X is convergent (in the norm) if there exists f in X || n − || { n} such that f converges in the norm to f. A Banach space is a normed linear space in { n} which every Cauchy sequence is convergent. Silly/important fact of the day: On February 2, 2011, MathSciNet lists 20331 publications with “Banach” in the title (and only 13539 publications with “Hilbert” in the title). 2 3 Examples of Banach spaces Examples of Banach spaces are given in propositions in this section. Proposition 3.1 For any p [1, ), Fn is a Banach space under the Lp-norm x = ∈ ∞ || ||p ( x p + + x p)1/p. Why is this a norm? H¨older’s inequality helps. For now I am | 1| ··· | n| assuming this, and I am not sure that I will get back to this. Presumably you have seen H¨older’s inequality. And you probably have seen that the open ball topology determined by this norm is equivalent to the standard Euclidean topology (with p =2), in which case, you should be able to prove easily that this is a Banach space. We will actually prove more general facts later. Definition 3.2 For any topological space X, let C(X) = f : X F,f continuous . { → } Note that if X is compact, then the uniform norm, or the sup norm, f = sup f(x) : x X || || {| | ∈ } is indeed a norm on C(X). If X is not compact, sup f(x) : x X may take on the value {| | ∈ } , so we cannot have a norm. ∞ Proposition 3.3 If X is a compact topological space, then C(X) is a Banach space. Proof. Let f be a Cauchy sequence in C(X). Then for all ǫ > 0, there exists N such { n} that for all m, n N, f f < ǫ. In particular, for all x X, f (x) f (x) ≥ || m − n|| ∈ | m − n | ≤ f f < ǫ. It follows that f (x) is a Cauchy sequence in F. Since F is complete, || m − n|| { n } we thus have that for all x X, there exists f(x) F such that f (x) f(x). So f ∈ ∈ n → { n} converges pointwise to a function f. We need to verify that f C(X). It suffices to prove that for every open set W ∈ in F, its f-preimage V in X is an open subset of X. For that it suffices to prove that for every x V there exists an open neighborhood U of x such that U V . So let x V . ∈ ⊆ ∈ Since W is open in a metric space, there exists ǫ > 0 such that B(f(x), ǫ) W . By the ⊆ Cauchy sequence assumption there exists N such that for all m, n N, f f < ǫ/3. ≥ || m − n|| In particular, for all y X, f (y) f (y) f f < ǫ/3, and for any m N, ∈ | m − n | ≤ || m − n|| ≥ fm(y) f(y) = fm(y) lim fn(y) = lim fm(y) fn(y) ǫ/3. | − | | − n→∞ | n→∞ | − | ≤ −1 SincefN is continuous, U = fN (B(fN (x), ǫ/3)) is an open neighborhood of x in X. For all y U, ∈ f(x) f(y) f(x) f (x) + f (x) f (y) + f (y) f(y) < ǫ/3 + ǫ/3 + ǫ/3 = ǫ, | − |≤| − N | | N − N | | N − | so that f(y) B(f(x), ǫ) W . Thus x U V , and U is open, as desired. ∈ ⊆ ∈ ⊆ 3 Finally, we need to prove convergence in the norm. For all m N (with N as above), ≥ f f = sup f (x) f(x) : x X || m − || {| m − | ∈ } = sup f (x) lim f (x) : x X {| m − n | ∈ } ǫ. ≤ And we’re done, without needing to use the Hausdorff condition. Definition 3.4 For any topological space X, define C (X) = f : X F continuous and bounded b { → } with the metric f = sup f(x) : x X . || || {| | ∈ } Proposition 3.5 For any topological space X, Cb(X) is a Banach space with the uniform norm. [If X is compact, then Cb(X) is a subset of the already established Banach space C(X).] Proof. It is straightforward to see that C (X) is a normed linear space. If f is a Cauchy b { n} sequence in C (X), then there exists B such that for all n, f B. Thus any pointwise b || n|| ≤ limit of f is also bounded. The rest of this proof is just like for C(X). { n} Definition 3.6 Let X be a topological space. A function f : X F is said to have → compact support if for all ǫ > 0, the set x : f(x) ǫ is compact. Define { | | ≥ } C (X) = f : X F continuous with compact support . 0 { → } Proposition 3.7 For any topological space X, C0(X) is a closed linear subspace of Cb(X), and hence a Banach space (under the uniform norm). Proof. We first show that C (X) C (X). Let f C (x). For all n, define S x X : 0 ⊆ b ∈ 0 n{ ∈ f(x) n . By assumption, S is compact, and since f is continuous, S is closed. Since | | ≥ } n n f is F-valued, S = . Thus by HW 1.3, there exists n such that S = , which means ∩ n ∅ n ∅ that f is bounded. Clearly C (X) is closed under scalar multiplication. Now let f, g C (X). Let ǫ > 0. 0 ∈ 0 Then x : f(x) + g(x) ǫ x : f(x) ǫ/2 x : g(x) ǫ/2 , { | | ≥ }⊆{ | | ≥ }∪{ | | ≥ } and since f + g is continuous, this says that x : f(x) + g(x) ǫ is a closed subset of { | | ≥ } a compact set, whence compact itself by Proposition 1, which proves that C0(X) is closed under addition. Since 0 C (X), it follows that C (X) is a vector subspace of C (X). ∈ 0 0 b We next prove that C (X) is a closed subset of C (X). If f is a Cauchy sequence 0 b { n} in C (X) in the norm, then f is a Cauchy sequence in C (X) in the norm. Since C (X) 0 { n} b b is Banach, there exists f C (X) such that f f in the norm. Let ǫ > 0. Then there ∈ b n → 4 exists N such that for all n N, f f < ǫ/2. If f(x) ǫ, then by the triangle ≥ || n − || | | ≥ inequality, ǫ f(x) f(x) f (x) + f (x) f f + f (x) < ǫ/2 + f (x) . ≤| |≤| − N | | N | ≤ || − N || | N | | N | Thus f (x) ǫ/2. Thus x : f(x) ǫ x : f (x) ǫ/2 .
Load more

Recommended publications
  • Chapter 4 the Lebesgue Spaces
    Chapter 4 The Lebesgue Spaces In this chapter we study Lp-integrable functions as a function space. Knowledge on functional analysis required for our study is briefly reviewed in the first two sections. In Section 1 the notions of normed and inner product spaces and their properties such as completeness, separability, the Heine-Borel property and espe- cially the so-called projection property are discussed. Section 2 is concerned with bounded linear functionals and the dual space of a normed space. The Lp-space is introduced in Section 3, where its completeness and various density assertions by simple or continuous functions are covered. The dual space of the Lp-space is determined in Section 4 where the key notion of uniform convexity is introduced and established for the Lp-spaces. Finally, we study strong and weak conver- gence of Lp-sequences respectively in Sections 5 and 6. Both are important for applications. 4.1 Normed Spaces In this and the next section we review essential elements of functional analysis that are relevant to our study of the Lp-spaces. You may look up any book on functional analysis or my notes on this subject attached in this webpage. Let X be a vector space over R. A norm on X is a map from X ! [0; 1) satisfying the following three \axioms": For 8x; y; z 2 X, (i) kxk ≥ 0 and is equal to 0 if and only if x = 0; (ii) kαxk = jαj kxk, 8α 2 R; and (iii) kx + yk ≤ kxk + kyk. The pair (X; k·k) is called a normed vector space or normed space for short.
    [Show full text]
  • Functional Analysis Lecture Notes Chapter 3. Banach
    FUNCTIONAL ANALYSIS LECTURE NOTES CHAPTER 3. BANACH SPACES CHRISTOPHER HEIL 1. Elementary Properties and Examples Notation 1.1. Throughout, F will denote either the real line R or the complex plane C. All vector spaces are assumed to be over the field F. Definition 1.2. Let X be a vector space over the field F. Then a semi-norm on X is a function k · k: X ! R such that (a) kxk ≥ 0 for all x 2 X, (b) kαxk = jαj kxk for all x 2 X and α 2 F, (c) Triangle Inequality: kx + yk ≤ kxk + kyk for all x, y 2 X. A norm on X is a semi-norm which also satisfies: (d) kxk = 0 =) x = 0. A vector space X together with a norm k · k is called a normed linear space, a normed vector space, or simply a normed space. Definition 1.3. Let I be a finite or countable index set (for example, I = f1; : : : ; Ng if finite, or I = N or Z if infinite). Let w : I ! [0; 1). Given a sequence of scalars x = (xi)i2I , set 1=p jx jp w(i)p ; 0 < p < 1; 8 i kxkp;w = > Xi2I <> sup jxij w(i); p = 1; i2I > where these quantities could be infinite.:> Then we set p `w(I) = x = (xi)i2I : kxkp < 1 : n o p p p We call `w(I) a weighted ` space, and often denote it just by `w (especially if I = N). If p p w(i) = 1 for all i, then we simply call this space ` (I) or ` and write k · kp instead of k · kp;w.
    [Show full text]
  • Elements of Linear Algebra, Topology, and Calculus
    00˙AMS September 23, 2007 © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. Appendix A Elements of Linear Algebra, Topology, and Calculus A.1 LINEAR ALGEBRA We follow the usual conventions of matrix computations. Rn×p is the set of all n p real matrices (m rows and p columns). Rn is the set Rn×1 of column vectors× with n real entries. A(i, j) denotes the i, j entry (ith row, jth column) of the matrix A. Given A Rm×n and B Rn×p, the matrix product Rm×p ∈ n ∈ AB is defined by (AB)(i, j) = k=1 A(i, k)B(k, j), i = 1, . , m, j = 1∈, . , p. AT is the transpose of the matrix A:(AT )(i, j) = A(j, i). The entries A(i, i) form the diagonal of A. A Pmatrix is square if is has the same number of rows and columns. When A and B are square matrices of the same dimension, [A, B] = AB BA is termed the commutator of A and B. A matrix A is symmetric if −AT = A and skew-symmetric if AT = A. The commutator of two symmetric matrices or two skew-symmetric matrices− is symmetric, and the commutator of a symmetric and a skew-symmetric matrix is skew-symmetric. The trace of A is the sum of the diagonal elements of A, min(n,p ) tr(A) = A(i, i). i=1 X We have the following properties (assuming that A and B have adequate dimensions) tr(A) = tr(AT ), (A.1a) tr(AB) = tr(BA), (A.1b) tr([A, B]) = 0, (A.1c) tr(B) = 0 if BT = B, (A.1d) − tr(AB) = 0 if AT = A and BT = B.
    [Show full text]
  • The Open Handbook of Formal Epistemology
    THEOPENHANDBOOKOFFORMALEPISTEMOLOGY Richard Pettigrew &Jonathan Weisberg,Eds. THEOPENHANDBOOKOFFORMAL EPISTEMOLOGY Richard Pettigrew &Jonathan Weisberg,Eds. Published open access by PhilPapers, 2019 All entries copyright © their respective authors and licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. LISTOFCONTRIBUTORS R. A. Briggs Stanford University Michael Caie University of Toronto Kenny Easwaran Texas A&M University Konstantin Genin University of Toronto Franz Huber University of Toronto Jason Konek University of Bristol Hanti Lin University of California, Davis Anna Mahtani London School of Economics Johanna Thoma London School of Economics Michael G. Titelbaum University of Wisconsin, Madison Sylvia Wenmackers Katholieke Universiteit Leuven iii For our teachers Overall, and ultimately, mathematical methods are necessary for philosophical progress. — Hannes Leitgeb There is no mathematical substitute for philosophy. — Saul Kripke PREFACE In formal epistemology, we use mathematical methods to explore the questions of epistemology and rational choice. What can we know? What should we believe and how strongly? How should we act based on our beliefs and values? We begin by modelling phenomena like knowledge, belief, and desire using mathematical machinery, just as a biologist might model the fluc- tuations of a pair of competing populations, or a physicist might model the turbulence of a fluid passing through a small aperture. Then, we ex- plore, discover, and justify the laws governing those phenomena, using the precision that mathematical machinery affords. For example, we might represent a person by the strengths of their beliefs, and we might measure these using real numbers, which we call credences. Having done this, we might ask what the norms are that govern that person when we represent them in that way.
    [Show full text]
  • “Strange” Limits
    c Gabriel Nagy “Strange” Limits Notes from the Functional Analysis Course (Fall 07 - Spring 08) Prerequisites: The reader is assumed to be familiar with the Hahn-Banach Theorem and/or the theory of (ultra)filter convergence. References to some of my notes will be added later. Notations. Given a non-empty set S, we denote by `∞(S) the vector space of all bounded R functions x : S → . On occasion an element x ∈ `∞(S) will also be written as an “S-tuple” R R x = (xs)s∈S. If S is infinite, we denote by Pfin(S) the collection of all finite subsets of S. Assuming S is infinite, for an element x = (x ) ∈ `∞(S), we can define the quantities s s∈S R lim sup xs = inf sup xs , F ∈P (S) S fin s∈SrF lim inf xs = sup inf xs . S s∈S F F ∈Pfin(S) r Definition. Suppose S is infinite. A limit operation on S is a linear map Λ : `∞(S) → , R R satisfying the inequalities: lim inf x ≤ Λ(x) ≤ lim sup x , ∀ x = (x ) ∈ `∞(S). (1) s s s s∈S R S S Comment. Of course, the quantities lim supS xs and lim infS xs can be defined for ar- bitrary “S-tuples,” in which case these limits might equal ±∞. In fact, this allows one to use S-tuples which might take infinite values. In other words, one might consider maps x : S → [−∞, ∞]. We will elaborate on his point of view later in this note. With the above terminology, our problem is to construct limit operations.
    [Show full text]
  • Linear Spaces
    Chapter 2 Linear Spaces Contents FieldofScalars ........................................ ............. 2.2 VectorSpaces ........................................ .............. 2.3 Subspaces .......................................... .............. 2.5 Sumofsubsets........................................ .............. 2.5 Linearcombinations..................................... .............. 2.6 Linearindependence................................... ................ 2.7 BasisandDimension ..................................... ............. 2.7 Convexity ............................................ ............ 2.8 Normedlinearspaces ................................... ............... 2.9 The `p and Lp spaces ............................................. 2.10 Topologicalconcepts ................................... ............... 2.12 Opensets ............................................ ............ 2.13 Closedsets........................................... ............. 2.14 Boundedsets......................................... .............. 2.15 Convergence of sequences . ................... 2.16 Series .............................................. ............ 2.17 Cauchysequences .................................... ................ 2.18 Banachspaces....................................... ............... 2.19 Completesubsets ....................................... ............. 2.19 Transformations ...................................... ............... 2.21 Lineartransformations.................................. ................ 2.21
    [Show full text]
  • Hahn Banach Theorem
    HAHN-BANACH THEOREM 1. The Hahn-Banach theorem and extensions We assume that the real vector space X has a function p : X 7→ R such that (1.1) p(ax) = ap(x), a ≥ 0, p(x + y) ≤ p(x) + p(y). Theorem 1.1 (Hahn-Banach). Let Y be a subspace of X on which there is ` such that (1.2) `y ≤ p(y) ∀y ∈ Y, where p is given by (1.1). Then ` can be extended to all X in such a way that the above relation holds, i.e. there is a `˜: X 7→ R such that ˜ (1.3) `|Y = `, `x ≤ p(x) ∀x ∈ X. Proof. The proof is done in two steps: we first show how to extend ` on a direction x not contained in Y , and then use Zorn’s lemma to prove that there is a maximal extension. If z∈ / Y , then to extend ` to Y ⊕ {Rx} we have to choose `z such that `(αz + y) = α`x + `y ≤ p(αz + y) ∀α ∈ R, y ∈ Y. Due to the fact that Y is a subspace, by dividing for α if α > 0 and for −α for negative α, we have that `x must satisfy `y − p(y − x) ≤ `x ≤ p(y + x) − `y, y ∈ Y. Thus we should choose `x by © ª © ª sup `y − p(y − x) ≤ `x ≤ inf p(y + x) − `y . y∈Y y∈Y This is possible since by subadditivity (1.1) for y, y0 ∈ R `(y + y0) ≤ p(y + y0) ≤ p(y + x) + p(y0 − x). Thus we can extend ` to Y ⊕ {Rx} such that `(y + αx) ≤ p(y + αx).
    [Show full text]
  • Basic Differentiable Calculus Review
    Basic Differentiable Calculus Review Jimmie Lawson Department of Mathematics Louisiana State University Spring, 2003 1 Introduction Basic facts about the multivariable differentiable calculus are needed as back- ground for differentiable geometry and its applications. The purpose of these notes is to recall some of these facts. We state the results in the general context of Banach spaces, although we are specifically concerned with the finite-dimensional setting, specifically that of Rn. Let U be an open set of Rn. A function f : U ! R is said to be a Cr map for 0 ≤ r ≤ 1 if all partial derivatives up through order r exist for all points of U and are continuous. In the extreme cases C0 means that f is continuous and C1 means that all partials of all orders exists and are continuous on m r r U. A function f : U ! R is a C map if fi := πif is C for i = 1; : : : ; m, m th where πi : R ! R is the i projection map defined by πi(x1; : : : ; xm) = xi. It is a standard result that mixed partials of degree less than or equal to r and of the same type up to interchanges of order are equal for a C r-function (sometimes called Clairaut's Theorem). We can consider a category with objects nonempty open subsets of Rn for various n and morphisms Cr-maps. This is indeed a category, since the composition of Cr maps is again a Cr map. 1 2 Normed Spaces and Bounded Linear Oper- ators At the heart of the differential calculus is the notion of a differentiable func- tion.
    [Show full text]
  • Diffusion with Nonlocal Boundary Conditions 3
    DIFFUSION WITH NONLOCAL BOUNDARY CONDITIONS WOLFGANG ARENDT, STEFAN KUNKEL, AND MARKUS KUNZE Abstract. We consider second order differential operators Aµ on a bounded, Dirichlet regular set Ω ⊂ Rd, subject to the nonlocal boundary conditions u(z)= u(x) µ(z,dx) for z ∈ ∂Ω. ZΩ + Here the function µ : ∂Ω → M (Ω) is σ(M (Ω), Cb(Ω))-continuous with 0 ≤ µ(z, Ω) ≤ 1 for all z ∈ ∂Ω. Under suitable assumptions on the coefficients in Aµ, we prove that Aµ generates a holomorphic positive contraction semigroup ∞ Tµ on L (Ω). The semigroup Tµ is never strongly continuous, but it enjoys the strong Feller property in the sense that it consists of kernel operators and takes values in C(Ω). We also prove that Tµ is immediately compact and study the asymptotic behavior of Tµ(t) as t →∞. 1. Introduction W. Feller [13, 14] has studied one-dimensional diffusion processes and their cor- responding transition semigroups. In particular, he characterized the boundary conditions which must be satisfied by functions in the domain of the generator of the transition semigroup. These include besides the classical Dirichlet and Neumann boundary conditions also certain nonlocal boundary conditions. Subsequently, A. Ventsel’ [31] considered the corresponding problem for diffusion problems on a domain Ω ⊂ Rd with smooth boundary. He characterized boundary conditions which can potentially occur for the generator of a transition semigroup. Naturally, the converse question of proving that a second order elliptic operator (or more generally, an integro-differential operator) subject to certain (nonlocal) boundary conditions indeed generates a transition semigroup, has recieved a lot of attention, see the article by Galakhov and Skubachevski˘ı[15], the book by Taira [30] and the references therein.
    [Show full text]
  • Riesz-Fredhölm Theory
    Riesz-Fredh¨olmTheory T. Muthukumar [email protected] Contents 1 Introduction1 2 Integral Operators1 3 Compact Operators7 4 Fredh¨olmAlternative 14 Appendices 18 A Ascoli-Arzel´aResult 18 B Normed Spaces and Bounded Operators 20 1 Introduction The aim of this lecture note is to show the existence and uniqueness of Fredh¨olmintegral operators of second kind, i.e., show the existence of a solution x of x − T x = y for any given y, in appropriate function spaces. 2 Integral Operators Let E be a compact subset of Rn and C(E) denote the space of complex valued continuous functions on E endowed with the uniform norm kfk1 = supx2E jf(x)j. Recall that C(E) is a Banach space with the uniform norm. 1 Definition 2.1. Any continuous function K : E × E ! C is called a con- tinuous kernel. Since K is continuous on a compact set, K is both bounded, i.e., there is a κ such that jK(x; y)j ≤ κ 8x; y 2 E and uniformly continuous. In particular, for each " > 0 there is a δ > 0 such that jK(x1; y) − K(x2; y)j ≤ " 8y 2 E whenever jx1 − x2j < δ. Example 2.1 (Fredh¨olmintegral operator). For any f 2 C(E), we define Z T (f)(x) = K(x; y)f(y) dy E where x 2 E and K : E × E ! R is a continuous function. For each " > 0, Z jT f(x1) − T f(x2)j ≤ jK(x1; y) − K(x2; y)jjf(y)j dy ≤ "kfk1jEj E whenever jx1 − x2j < δ.
    [Show full text]
  • Embedding Banach Spaces
    Embedding Banach spaces. Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, and Andras Zsak January 24, 2009 Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. Definition (Banach space) A normed vector space (X ; k · k) is called a Banach space if X is complete in the norm topology. Definition (Schauder basis) 1 A sequence (xi )i=1 ⊂ X is called a basis for X if for every x 2 X there exists a 1 P1 unique sequence of scalars (ai )i=1 such that x = i=1 ai xi . Question (Mazur '36) Does every separable Banach space have a basis? Theorem (Enflo '72) No. Background Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. Definition (Schauder basis) 1 A sequence (xi )i=1 ⊂ X is called a basis for X if for every x 2 X there exists a 1 P1 unique sequence of scalars (ai )i=1 such that x = i=1 ai xi . Question (Mazur '36) Does every separable Banach space have a basis? Theorem (Enflo '72) No. Background Definition (Banach space) A normed vector space (X ; k · k) is called a Banach space if X is complete in the norm topology. Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. Question (Mazur '36) Does every separable Banach space have a basis? Theorem (Enflo '72) No. Background Definition (Banach space) A normed vector space (X ; k · k) is called a Banach space if X is complete in the norm topology. Definition (Schauder basis) 1 A sequence (xi )i=1 ⊂ X is called a basis for X if for every x 2 X there exists a 1 P1 unique sequence of scalars (ai )i=1 such that x = i=1 ai xi .
    [Show full text]
  • Nonlinear Spectral Theory Henning Wunderlich
    Nonlinear Spectral Theory Henning Wunderlich Dr. Henning Wunderlich, Frankfurt, Germany E-mail address: [email protected] 1991 Mathematics Subject Classification. 46-XX,47-XX I would like to sincerely thank Prof. Dr. Delio Mugnolo for his valuable advice and support during the preparation of this thesis. Contents Introduction 5 Chapter 1. Spaces 7 1. Topological Spaces 7 2. Uniform Spaces 11 3. Metric Spaces 14 4. Vector Spaces 15 5. Topological Vector Spaces 18 6. Locally-Convex Spaces 21 7. Bornological Spaces 23 8. Barreled Spaces 23 9. Metric Vector Spaces 29 10. Normed Vector Spaces 30 11. Inner-Product Vector Spaces 31 12. Examples 31 Chapter 2. Fixed Points 39 1. Schauder-Tychonoff 39 2. Monotonic Operators 43 3. Dugundji and Quasi-Extensions 51 4. Measures of Noncompactness 53 5. Michael Selection 58 Chapter 3. Existing Spectra 59 1. Linear Spectrum and Properties 59 2. Spectra Under Consideration 63 3. Restriction to Linear Operators 76 4. Nonemptyness 77 5. Closedness 78 6. Boundedness 82 7. Upper Semicontinuity 84 Chapter 4. Applications 87 1. Nemyckii Operator 87 2. p-Laplace Operator 88 3. Navier-Stokes Equations 92 Bibliography 97 Index 101 3 Introduction The term Spectral Theory was coined by David Hilbert in his studies of qua- dratic forms in infinitely-many variables. This theory evolved into a beautiful blend of Linear Algebra and Analysis, with striking applications in different fields of sci- ence. For example, the formulation of the calculus of Quantum Mechanics (e.g., POVM) would not have been possible without such a (Linear) Spectral Theory at hand.
    [Show full text]