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Links
Arzelà–Ascoli theorem, Banach space, Besov space, Compact operator, Compact operator on Hilbert space, Compact space, Convergence of Fourier series, Convolution, Distortion problem, Dual space, Fourier series,

Haar wavelet, Hahn-Banach theorem, Hardy space, Hilbert space, Hölder's inequality,

Interpolation space, Lorentz space, Lp space, Maximal function, Modulus and characteristic of convexity,

Reflexive space, Riemann series theorem, Riesz–Fischer theorem,

Schauder basis, Schur's property, Sequence space, Series (mathematics), Sobolev space,

Tsirelson space, Unconditional convergence, Uniformly convex space, Uniformly smooth space,

Vector measure, Vitali covering lemma, Weak topology

Introduction
Functional analysis aims to find functions that are solutions of various equations, several arising from physics. Abstract solutions, namely, functions that cannot be expressed by an explicit formula, are often obtained as limits in a well chosen vector space of functions X of a sequence of approximate solutions. Completeness of X is needed in order to make sure that the limit exists in X. Many examples of such spaces X, but not all, are Banach spaces.

Various type of compact sets in function spaces (norm compact, weak compact) are also used to prove the existence of abstract solutions, for example to optimization problems. In this respect, it is important to characterize compact subsets of function spaces.

A notation
It is sometimes convenient to denote the pairing between a linear functional f and a vector x by


 * $$(x, f) := f(x).$$

This pairing is bilinear, as opposed to inner products, that are antilinear in the second variable for complex scalars.

Adjoint operator
When T is a bounded linear operator from a normed space X to a normed space Y, the adjoint operator T&thinsp;′ (or transpose) is the linear map from the dual Y&thinsp;′ to the dual X&thinsp;′ defined by


 * $$ g \in Y' \ \mapsto \ T'(g) = g \circ T \in X'.$$

If the pairing between a linear functional f and a vector x is denoted by


 * $$(f, x) := f(x), \ \ \text{then:} \ \ \ (T'(g), x) = (g, T(x)), \quad g \in Y', \ x \in X.$$

One has that


 * $$T' \in B(Y', X'), \ \ \|T'\|_{B(Y', X')} = \|T\|_{B(X, Y)},

\ \ (\lambda T_1 + T_2)' = \lambda T_1' + T_2'.$$

The adjoint of the identity map in B(X) is the identity map. If U &isin; B(Y, Z) is given, then


 * $$(U \circ T)' = T' \circ U'.$$

It follows that the adjoint of an invertible operator T is invertible, with an inverse equal to the adjoint of T&thinsp;&minus;1. When two spaces are isomorphic, their duals are isomorphic. The converse is not true: for every countably infinite compact Hausdorff space K, the dual of C(K) is isometrically isomorphic to ℓ1, but these C(K) spaces need not be isomorphic (see details below).

The adjoint map T&thinsp;′ is continuous from the weak*-topology of Y&thinsp;′ to that of X&thinsp;′. Conversely, a linear map from Y&thinsp;′ to X&thinsp;′, continuous between the weak*-topologies, is the adjoint of a bounded linear map from X to Y.

If T is a bounded linear operator from a Hilbert space H1 to another Hilbert space H2, the Hilbertian adjoint T* is the bounded linear operator acting from H2 to H1 that is defined by


 * $$ \langle T^* (x_2), x_1 \rangle = \langle x_2, T(x_1) \rangle, \quad x_1 \in H_1, \ \ x_2 \in H_2.$$

If jH denotes the antilinear isometry from a Hilbert space H onto its dual defined by


 * $$ j_H(y)(x) = \langle x, y \rangle, \ \ x, y \in H,\ \ \text{then} \ \ T^* = j_{H_1}^{-1} \circ T' \circ j_{H_2}.$$

Dual inequalities
Considering the adjoint is a special instance of using the convex conjugate function. Several inequalities in Analysis have a dual form obtained by passing to the adjoint.

Compactness in Banach spaces
Norm- and weak-compactness are the two main notions of compactness in a general Banach space X, together with weak*-compactness that applies only in a dual space X = Y'.

Weak and weak*-compactness
In a weakly compact (resp. weak*-compact) set K, any decreasing chain of non-empty weakly closed (resp. weak*-closed) subsets has, by definition of compactness, a non-empty intersection. The intersection property is used in the proof of the Krein–Milman theorem, in order to exhibit extreme points of K as minimal faces in weak- or weak*-compact convex sets. Several tasks in convex analysis, in particular establishing upper bounds for convex functions, are done by reducing the question to the case of extreme points. Lindenstrauss' proof of Lyapounov theorem on vector measures is another example, relying on the identification of extreme points of a certain weak* compact convex set of functions in L&infin;([0, 1]).

Weak compactness is often used within a context of convexity, because norm-closed convex sets are weakly closed. This implies the existence of minima for continuous, or lower semi-continuous, convex real functions defined on a non-empty weakly compact convex set.

In non-reflexive spaces the structure of weakly compact subsets can be of special interest. In L1([0, 1]), weakly compact subsets are uniformly integrable. Conversely, a closed convex subset in L1 that is not weakly compact contains a basic sequence equivalent to the unit vector basis in ℓ1. In ℓ1, weakly compact sets are norm compact. This means that ℓ1 satisfies Schur's property.

A bounded linear operator between two Banach spaces X and Y is weakly compact if the image of the unit ball of X is relatively weakly compact in Y. If X is reflexive, the image by T &isin; B(X, Y) of the closed unit ball in X is weakly compact in Y, since T is also weakly continuous. More generally, T is weakly compact whenever it factors through a reflexive space. Conversely:

Theorem. A bounded linear operator between two Banach spaces is weakly compact if and only if it factors through a reflexive space.

Norm compactness
Riesz's lemma asserts that a normed space with compact unit ball is finite dimensional. This implies that eigenspaces of compact linear operators, for each non-zero eigenvalue, are finite dimensional. It is a first step in the direction of the rich Riesz's spectral theory of compact operators. The Schauder fixed point theorem extends the Brouwer fixed-point theorem to infinite dimensional settings. It is useful for proving existence of solutions to nonlinear partial differential equations. Lomonosov's theorem on the existence of invariant subspaces can be proved by a nice application of Schauder's fixed point theorem.

Various criteria characterize compact subsets of Banach spaces. A general lemma of Grothendieck asserts that a closed set K is norm compact in a Banach space X if and only if it is contained in the closed convex hull of a sequence of vectors tending to 0 in X,
 * $$ K \subset \overline{\operatorname{conv}} \{ x_n\}_{n=1}^\infty, \ \ \{ x_n\}_{n=1}^\infty \subset X, \ \ \lim_n \|x_n\|_X = 0.$$

More applicable criteria characterize compact subsets in specific function spaces: the Ascoli theorem characterizes relatively compact subsets in C(K). It implies in particular that the adjoint of a compact linear operator is again compact. It also proves compactness of specific operators, such as the Volterra operator, and many similar examples in the theory of Volterra integral equations. Compact subsets in Lp are characterized by the Fréchet-Kolmogorov theorem. This characterization permits to understand compact Sobolev embeddings.

Banach space theory
Although functional analysis primarily looks for methods to find functions that are solutions of equations from physics, Banach's book as started the study of Banach spaces "for themselves". Lindenstrauss, V. Milman around 1970 gave new impetus to this direction of research.

One direction is the search for "nice structures" in every Banach space. For example, find "nice" infinite dimensional subspaces, with a basis (a result stated in Banach's book), existence of a c_0 or l^p subspace, or of an unconditional basic sequences. Negative solutions for most questions about infinite dimensional nice subspaces.

Classify, characterize classes of Banach spaces. Some properties only depend upon two dimensional subspaces: for example, the parallelogram law characterizing Hilbert space. From the parallelogram law, cutting equality in two inequalities and changing them to
 * $$ \|x+y\|^p + \|x-y\|^p \le \|x\|^p + C^p \|y\|^p, \ \ \|x+y\|^q + \|x-y\|^q \ge \|x\|^q + c^q \|y\|^q$$

gives particular cases of uniform smoothness and uniform convexity respectively, again depending only upon two dimensional subspaces. Replacing the average over two signs by the study of the more general expression
 * $$ \operatorname{Ave}_{\pm} \sum_{j=1}^n \pm x_j$$

leads to the definition of spaces of type p and cotype q. Kwapien's theorem characterizes spaces isomorphic to Hilbert spaces as being those that have both type 2 and cotype 2.

Type and cotype do not depend only on two dimensional subspaces, but they only depend on the family of finite dimensional subspaces of X. Superproperties introduced by James. Superreflexive, James, Enflo, trees in X are special instances of X-valued martingales, Pisier's renorming uses martingales inequalities.

No hope for classific of subspaces, even in a space simple as l^p (contains a couterexample to the approximation problem). But sometimes complemented subspaces of a given space can be completely characterized, l^p and decomposition method, L^1 still unknown, primary spaces like L^p

Construct spaces; by hand; by interpolation theory

distortion in l^2

operators: are they enough operators, approx, invariant subspace in l^2, lambda i + k

Operator spaces

Local theory
Dvoretzky's theorem

random methods, Kashin, Gluskin used to show chaotic behavior.

Krivine

Milman's ellipsoid, Pisier's book, inverse Santalo

=== Non linear theory Benyamini-Lindenstrauss

Handbook, Tomczak, Wojtaszczyk

To do
Read "Reflexive space"

Notation (x ', x) for duality

Banach space: def. weakly compact operator

Reflexive space: ell^2 sum of reflexive spaces

Great Baire theorem in Baire function

adjoint map

Correct Bessel spaces in Sobolev space

L^p(L^q) interpolation (see Bergh-L)

complete Besov article

"second dual", "w*" --> weak*

Dvoretzky–Rogers

Classification of complemented subspaces, $$\ell_p$$, Sobczyk

Holomorphic families on the circle, families of operators

According to Lions–Peetre, abstract Marcinkiewicz. Reiteration and Marcinkiewicz

Duality with Calderon

Benedek Panzone L^p(L^q)

Quasinorm and Rolewicz

Lorentz norm with f**

Wojtaszczyk, Wavelets,

equi-integrability does not exist in WP? uniform integrability but infinite measure to do

cosine series in Fourier series

Lorentz spaces

Wavelet bases (Woj's book), bases in H^1,

Birnbaum–Orlicz space Orlicz sequences spaces

give page in Bessaga–Pelczynski

Franklin is unconditional in L^p

boundedly complete and unconditional

Done
(done) pairing for dual of c_0

(done) Baire property with Odell-Rosenthal, link to Baire function

(done) Interpolation for Sobolev and Besov, interpolation and Besov spaces

(done) norming functional in Banach space

(done) Odell-Rosenthal in Banach space

(done) Create bidual section after examples of dual, move material from "Reflexivity", discuss example of c_0, add Goldstine theorem

(done) adding to refs: Bergh and Löfström, (1976), Interpolation Spaces: An Introduction, Springer-Verlag,

Ryan, Raymond A. (2000), Introduction to Tensor Products of Banach Spaces, Springer-Verlag

(done) modulus article, equivalent to convex function

(done) super-reflexive in Reflexive space and links

(done) Uniformly smooth/convex, Asplund

(done) Weak topology is Hausdorff

(done) Change words on polarization

(done) relies on Dvoretzky's theorem, more on homogeneous (Komorowski)

(done) Hilbert and Banach spaces: changing the section with figure, include Kwapien, Lindenstrauss–Tzafriri and homogeneous space problem

(done) sequences: Rosenthal $$\ell^1$$ theorem

(done) Complete Megginson in Banach space and correct inline refs accordingly

(done) Say: "linear span"

(done) total and HB

(done) Correct and complete Ryan's ref in Banach space

(done) Basis for K(X)

(done) A(D) is example 10 p. 12 in Banach

(done in "Banach space") Complemented subspaces?

(done with Carothers) link to Johnson-Lindenstrauss about every separable is quotient of ell^1, HB vol 1, p. 17

(done) Discrete, abstract interpolation spaces, factorization of weakly compact operators

(done) Reiteration theorem

(done) Group K- and J-exact interpolation pairs

(done) copy Golubov in Haar wavelet

(done) questions A(D): page in Banach, p. 238, &sect;3

(done) mention interpolation space at Marcinkiewicz interpolation theorem and Riesz–Thorin theorem

Symbols
〈•,•〉 ℓ1  nowiki: { , norm with one symbol: ǁxǁ

MOS:MATH

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