Notes on Functional Analysis

Frontmatter

Lecture 1. Banach Spaces

Abstract

The subject Functional Analysis was created at the beginning of the twentieth century to provide a unified framework for the study of problems that involve continuity and linearity. The basic objects of study in this subject are Banach spaces and linear operators on these spaces.

Rajendra Bhatia

Lecture 2. Dimensionality

Abstract

Let X be a vector space and let S be a subset of it. We say S is linearly independent if for every finite subset {x₁,…, x _n } of S, the equation

$${a_1}{x_1} + \cdots + {a_n}{x_n} = 0$$

(2.1)

holds if and only if a₁ = a₂ = ⋯ = a_n = 0. A (finite) sum like the one in (2.1) is called a linear combination of x₁,…, x_n.

Rajendra Bhatia

Lecture 3. New Banach Spaces from Old

Abstract

Let X be a vector space and M a subspace of it. Say that two elements x and y of X are equivalent, x ~ y, if x − y ∈ M. This is an equivalence relation on X. The coset of x under this relation is the set

$$\tilde x = x + M: = \left\{ {x + m:m \in M} \right\}.$$

. Let \(\tilde X\) be the collection of all these cosets. If we set

$$\begin{array}{*{20}{c}} {\tilde x + \tilde y = \widetilde{x + y},} \\ {\alpha \tilde x = \widetilde{\alpha x},} \\ \end{array}$$

then \(\tilde X\) is a vector space with these operations.

Rajendra Bhatia

Lecture 4. The Hahn-Banach Theorem

Abstract

The Hahn-Banach Theorem (H.B.T.) is called one of the three basic principles of linear analysis—the two others are the Uniform Boundedness Principle and the Open Mapping Theorem. We will study them in the next three lectures. The H.B.T. has several versions and several corollaries.

Rajendra Bhatia

Lecture 5. The Uniform Boundedness Principle

Abstract

The Baire Category Theorem says that a complete metric space cannot be the union of a countable number of nowhere dense sets. This has several very useful consequences. One of them is the Uniform Boundedness Principle (U.B.P.) also called the Banach-Steinhaus Theorem.

Rajendra Bhatia

Lecture 6. The Open Mapping Theorem

Abstract

Theorems that tell us that a continuous map is also open under some simple conditions play a very important role in analysis. The open mapping theorem is one such result.

Rajendra Bhatia

Lecture 7. Dual Spaces

Abstract

The idea of duality, and the associated notion of adjointness, are important in functional analysis. We will identify the spaces X* for some of the standard Banach spaces.

Rajendra Bhatia

Lecture 8. Some Applications

Abstract

Let μ _n be a sequence of probability measures on [0, 1]. Then there exists a subsequence μ _m and a probability measure μ such that

$$\int {fd\mu \to \int {fd\mu \quad as\; \to \infty } }$$

for all f ∈ C[0, 1].

Rajendra Bhatia

Lecture 9. The Weak Topology

Abstract

When we say that a sequence f _n in the space C[0, 1] converges to f, we mean that ‖f _n − f‖ → 0 as n → ∞; and this is the same as saying f _n converges to f uniformly. There are other notions of convergence that are weaker, and still very useful in analysis. This is the motivation for studying different topologies on spaces of functions, and on general Banach spaces.

Rajendra Bhatia

Lecture 10. The Second Dual and the Weak* Topology

Abstract

The dual of X* is another Banach space X**. This is called the second dual or the bidual of X. Let J be the map from X into X** that associates with x ∈ X the element F _x ∈ X** defined as

$${F_x}(f) = f(x)\;for\;all\;f \in {X^*}.$$

Then J is a linear map and ‖Jx‖ = ‖x‖. (See (9.2).) Thus J is an isometric imbedding and we can regard X as a subspace of X**.

Rajendra Bhatia

Lecture 11. Hilbert Spaces

Abstract

To each vector in the familiar Euclidean space we assign a length, and to each pair of vectors an angle between them. The first notion has been made abstract in the definition of a norm. What is missing in the theory so far is an appropriate concept of angle and the associated notion of orthogonality. These ideas depend on the introduction of an inner product Hilbert spaces are special kinds of Banach spaces whose norms arise from inner products.

Rajendra Bhatia

Lecture 12. Orthonormal Bases

Abstract

A subset E in a Hilbert space is said to be an orthonormal set if 〈e₁, e₂〉 = 0 for all e₁, e₂ in E (e₁ ∈ e₂), and ‖e‖ = 1 for all e in E.

Rajendra Bhatia

Lecture 13. Linear Operators

Abstract

Let X, Y be Banach spaces. For a while we will study bounded linear operators from X to Y. These will just be called operators.

Rajendra Bhatia

Lecture 14. Adjoint Operators

Abstract

Every operator A from X to Y gives rise, in a natural way to an operator A* from the dual space Y* to X*. Many properties of A can be studied through this operator called the adjoint of A.

Rajendra Bhatia

Lecture 15. Some Special Operators in Hilbert Space

Abstract

The additional structure in a Hilbert space and its self-duality make the adjoint operation especially interesting. All Hilbert spaces that we consider are over complex scalars except when we say otherwise.

Rajendra Bhatia

Lecture 16. The Resolvent and The Spectrum

Abstract

A large, and the most important, part of operator theory is the study of the spectrum of an operator. In finite dimensions, this is the set of eigenvalues of A. In infinite dimensions there are complications that arise from the fact that an operator could fail to be invertible in different ways. Finding the spectrum is not an easy problem even in the finite-dimensional case; it is much more difficult in infinite dimensions.

Rajendra Bhatia

Lecture 17. Subdivision of the Spectrum

Abstract

Let S be the right shift operator on the space ℓ₁. Since ‖S‖ = 1 the spectrum σ(S) is contained in the closed unit disk D. We have seen that S has no eigenvalue. The adjoint of S is the left shift operator T on the space ℓ_∞. If λ is any complex number with |λ| ≤ 1, then the vector x_λ = (1, λ, λ²,…) is in ℓ_∞ and Tx_λ = λx_λ. Thus every point λ in the disk D is an eigenvalue of T. This shows also that σ(S) = σ(T) = D.

Rajendra Bhatia

Lecture 18. Spectra of Normal Operators

Abstract

In Lecture 15 we studied normal operators in Hilbert spaces. For this class the spectrum is somewhat simpler.

Rajendra Bhatia

Lecture 19. Square Roots and the Polar Decomposition

Abstract

One of the most important and useful theorems of linear algebra is the spectral theorem. This says that every normal operator on an n-dimensional Hilbert space ℋ can be diagonalised by a unitary conjugation: there exists a unitary operator U such that U*AU = Λ, where Λ is the diagonal matrix with the eigenvalues of A on its diagonal. Among other things, this allows us to define functions of a normal matrix A in a natural way. Let f be any functions on ℂ. If Λ = diag (λ₁,…, λ _n ) is a diagonal matrix with λ _j as its diagonal entries, define f(Λ) to be the diagonal matrix diag (f(λ₁),…,f(λ _n )), and if A = UΛU*, put f(A) = Uf(Λ)U*.

Rajendra Bhatia

Lecture 20. Compact Operators

Abstract

This is a special class of operators and for several reasons it is good to study them in some detail at this stage. Their spectral theory is much simpler than that of general bounded operators, and it is just a little bit more complicated than that of finite-dimensional operators. Many problems in mathematical physics lead to integral equations, and the associated integral operators are compact. For this reason these operators were among the first to be studied, and in fact, this was the forerunner to the general theory.

Rajendra Bhatia

Lecture 21. The Spectrum of a Compact Operator

Abstract

Most of the spectral properties of a compact operator in a Banach space were discovered by F. Riesz, and appeared in a paper in 1918 (several years before Banach’s book). These results were augmented and simplified by the work of Schauder. What follows is an exposition of these ideas.

Rajendra Bhatia

Lecture 22. Compact Operators and Invariant Subspaces

Abstract

Continuing the analysis of the previous lecture we obtain more information about compact operators.

Rajendra Bhatia

Lecture 23. Trace Ideals

Abstract

Let A be a compact operator on (an infinite-dimensional) Hilbert space ℋ and let

$${s_1}\left( A \right) \geqslant {s_2}\left( A \right) \geqslant \cdots \geqslant 0$$

(23.1)

be the singular values of A. The sequence s _n (A) converges to 0. In this lecture we study special compact operators for which this sequence belongs to the space ℓ₁ or the space ℓ₂.

Rajendra Bhatia

Lecture 24. The Spectral Theorem -I

Abstract

Let A be a Hermitian operator on the space ℂ ⁿ . Then there exists an orthonormal basis {e _j } of ℂ ⁿ each of whose elements is an eigenvector of A. We thus have the representation

$$A = \sum\limits_{j = 1}^n {{\lambda _j}\left\langle { \cdot ,{e_j}} \right\rangle {e_j},}$$

(24.1)

where Ae _j = λ _j e _j . We can express this in other ways. Let λ₁ > λ₂ > ⋯ > λ _k be the distinct eigenvalues of A and let m₁, m₂,&, m _k be their multiplicities. Then there exists a unitary operator U such that

$$U*AU = \sum\limits_{j = 1}^k {{\lambda _j}{P_j},}$$

(24.2)

, where P₁, P₂,&, P _k are mutually orthogonal projections and

$$\sum\limits_{j = 1}^k {{P_j} = I.}$$

(24.3)

. The range of P _j , is the m _j -dimensional eigenspace of A corresponding to the eigenvalue λ _j . This is called the spectral theorem for finite-dimensional operators.

Rajendra Bhatia

Lecture 25. The Spectral Theorem -II

Abstract

Look at the formulation of the finite-dimensional spectral theorem given in (24.2) and (24.3). For infinite dimensional compact operators the finite sum is replaced by an infinite sum. Going further we may replace the sum by an integral. The second version of the spectral theorem says that every self-adjoint operator can be represented by such an integral. The integration is now with respect to a projection-valued measure (instead of an ordinary positive measure) and the resulting definite integral is a self-adjoint operator (instead of a number.)

Rajendra Bhatia

Lecture 26. The Spectral Theorem -III

Abstract

This lecture is a quick review of some matters related to the spectral theorem.

Rajendra Bhatia

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