A Friendly Approach to Functional Analysis

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Amol Sasane

• Publisher: WSPC (EUROPE)
• Year: 2017
• ISBN:
  9781786343369
• Topic: Mathematik
• Subtopic: Funktionsanalyse

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This book constitutes a concise introductory course on Functional Analysis for students who have studied calculus and linear algebra. The topics covered are Banach spaces, continuous linear transformations, Frechet derivative, geometry of Hilbert spaces, compact operators, and distributions. In addition, the book includes selected applications of functional analysis to differential equations, optimization, physics (classical and quantum mechanics), and numerical analysis. The book contains 197 problems, meant to reinforce the fundamental concepts. The inclusion of detailed solutions to all the exercises makes the book ideal also for self-study.

A Friendly Approach to Functional Analysis is written specifically for undergraduate students of pure mathematics and engineering, and those studying joint programmes with mathematics.

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Chapter 1

Normed and Banach spaces

As we had discussed in the introduction, we wish to do calculus in vector spaces (such as C[a, b], whose elements are functions). In order to talk about the concepts from calculus such as differentiability, we need a notion of closeness between points of a vector space.

Recall for example, that a real sequence (a_n)_n∈N is said to converge with limit L ∈ R if for every

> 0, there exists an N ∈ N such that whenever n > N, |a_n − L| <

. In other words, the sequence converges to L if no matter what distance

> 0 is given, one can guarantee that all the terms of the sequence beyond a certain index N are at a distance of at most

away from L (this is the inequality |a_n − L| <

). So we notice that in this notion of “convergence of a sequence”, indeed the notion of distance played a crucial role. After all, we want to say that the terms of the sequence get “close” to the limit, and to measure closeness, we use the distance between points of R. A similar thing happens with continuity and differentiability. Recall that a function f : R → R is said to be continuous at c ∈ R if for every

> 0, there exists a δ > 0 such that whenever |x − c| < δ, |f(x) − f(c)| <

. Roughly, given any distance

, I can find a distance δ such that whenever I choose an x not farther than a distance δ from c, I am guaranteed that f(x) is not farther than a distance of

from f(c). Again notice the key role played by the distance in this definition. The distance between points x, y ∈ R is taken as |x − y|, where | · | : R → [0, ∞) is the absolute value function, given by

If we imagine the real numbers depicted on a “number line”, then |x − y| is the length of line segment joining x, y visualised on the number line. See the following picture.

But now if one wants to also do calculus in a vector space X (for example C[a, b]), there is so far no ready-made available notion of distance between vectors. One way of creating a distance in a vector space is to equip it with a...