Introductory Functional Analysis with Applications: 17 (Wiley Classics Library)

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Observe the notation; we write Tx instead of T(x); this simplification is standard in functional analysis. Furthermore, for the remainder

each of these sequences be the center of a small ball, say, of radius 1/3, these balls do not intersect and we have uncountably many of them. If M is any dense set in I"", each of these nonintersecting balls must contain an element of M. Hence M cannot be countable. Since M was an arbitrary dense set, this shows that 1 cannot have dense subsets which are countable. Consequently, 1 is not separable. 00 This metric space (X, d) is not complete. In fact, an example of a Cauchy sequence without limit in X is given by any sequence of polynomials which converges uniformly on J to a continuous function, not a polynomial. If (x,.) is Cauchy and has a convergent subsequence, say, show that (x,.) is convergent with the limit x. Spectral Properties of Bounded Self-Adjoint Linear Operators 460 9.2 Further Spectral Properties of Bounded Self-Adjoint Linear Operators 465 9.3 Positive Operators 469 9.4 Square Roots of a Positive Operator 476 9.5 Projection Operators 480 9.6 Further Properties of Projections 486 9.7 Spectral Family 492 9.8 Spectral Family of a Bounded Self-Adjoint Linear Operator 497 9.9 Spectral Representation of Bounded Self-Adjoint Linear Operators 505 9.10 Extension of the Spectral Theorem to Continuous Functions 512 9.11 Properties of the Spectral Family of a Bounded SelfAd,ioint Linear Operator 516Show that for r = n, the norm in Prob. 14 is the natural norm corresponding to 11·lb and II· liz as defined in that problem. Definite iiltegral. The definite integral is a number if we consider it for a single function, as we do in calculus most of the time. However, the situation changes completely if We consider that integral for all functions in a certain function space. Then the integral becomes a functional on that space, call it f As a space let us choose C[a, b]; cf. 2 ..2-5. Then I is defined by

Normed Space. Banach Space The examples in the last section illustrate that in many cases a vector space X may at the same time be a metric space because a metric d is defined on X. However, if there is no relation between the algebraic structure and the metric, we cannot expect a useful and applicable theory that combines algebraic and metric concepts. To guarantee such a relation between "algebraic" and "geometric" properties of X we define on X a metric d in a special way as follows. We first introduce an auxiliary concept, the norm (definition below), which uses the algebraic operations of vector space. Then we employ the norm to obtain a metric d that is of the desired kind. This idea leads to the concept of a normed space. It turns out that normed spaces are special enough to provide a basis for a rich and interesting theory, but general enough to include many concrete models of practical importance. In fact, a large number of metric spaces in analysis can be regarded as normed spaces, so that a normed space is probably the most important kind of space in functional analysis, at least from the viewpoint of present-day applications. Here are the definitions: 2.2-1 Definition (Normed space, Banach space). A normed space 3 X is a vector space with a norm defined on it, A Banach space is a 3 Also called a normed vector space or normed linear space. The definition was given (independently) by S. Banach (1922), H. Hahn (1922) and N. Wiener (1922). The theory developed rapidly, as can be seen from the treatise by S. Banach (1932) published only ten years later. Chapter.2.4-2.10 Marián Fabian,Petr Habala,Petr Hájek, Vicente Montesinos,Václav Zizler, Banach Space Theory, The Basis for Linear and Nonlinear Analysis Section 1.3 V. Hence if M is dense in X, then every ball in X, no matter how small, will contain points of M; or, in other words, in this case there is no point x E X which has a neighborhood that does not contain points of M. We shall see later that separable metric spaces are somewhat simpler than nonseparable ones. For the time being, let us consider some important examples of separable and nonseparable spaces, so that we may become familiar with these basic concepts. In [00, let Y be the subset of all sequences with only finitely many nonzero terms. Show that Y is a subspace of [00 but not a closed subspace. 4. (Continuity of vector space operations) Show that in a normed space X, vector addition and multiplication by scalars are continuous operations with respect to the norm; that is, the mappings defined by (x, y) ~ x+y and (a, x) ~ ax are continuous. 5. Show that x" ~ x and Yn ~ Y implies Xn + Yn an ~ a and Xn ~ x implies anx" ~ ax. The Wiley Classics Library consists of selected books originally published by John Wiley & Sons that have become recognized classics in their respective fields. With these new unabridged and inexpensive editions, Wiley hopes to extend the life of these important works by making them available to future generations of mathematicians and scientists.Show that all complex m x n matrices A = (ajk) with fixed m and n constitute an mn-dimensional vector space Z. Show that all norms on Z are equivalent. What would be the analogues of II· III> I . 112 and I . 1100 in Prob. 8, Sec. 2.2, for the present space Z? In a finite dimensional normed space the closed unit ball is compact by Theorem 2.5-3. Conversely, Riesz's lemma gives the following useful and remarkable 2.5-5 Theorem (Finite dimension). If a normed space X has the property that the closed unit ball M = {x Illxll ~ I} is compact, then X is finite dimensional. cannot be obtained from a norm. This may immediately be seen from the following lemma which states two basic properties of a metric d obtained from a norm. The first property, as expressed by (9a), is called the translation invariance of d. 2.2-9 Lemma (Translation invariance). . on a normed space X satisfies The SUbscript x is a little reminder that we got g by the use of a certain x E X. The reader should observe carefully that here I is the variable whereas x is fixed. Keeping this in mind, he should not have difficulties in understanding our present consideration. &, as defined by (4) is linear. This can be seen from

i) Construct an element x (to be used as a limit). (ii) Prove that x is in the space considered. (iii) Prove convergence Xn ~ x (in the sense of the metric). This is a review for Wiley and the publisher taking the Indian market for granted. It is not for the content of this book, or the author. Show that x,. --- x if and only if for every neighborhood V of x there is an integer no such that Xn E V for all n > no. 4. (Boundedness) Show that a Cauchy sequence is bounded. 5. Is boundedness of a sequence in a metric space sufficient for the sequence to be Cauchy? Convergent? 6. If (x,.) and (Yn) are Cauchy sequences in a metric space (X, d), show that (an), where an = d(x,., Yn), converges. Give illustrative examples. 7. Give an indirect proof of Lemma 1.4-2(b). 8. If d 1 and d 2 are metrics on the same set X and there are positive numbers a and b such that for all x, YE X, ad 1 (x, y);a d 2 (x, y);a bd 1 (x, Y), these cosets constitute the elements of a vector space. This space is called the quotient space (or sometimes factor space) of X by Y (or modulo Y) and is denoted by X/Yo Its dimension is called the codimension of Y and is denoted by codim Y, that is, codim Y=dim(X/Y).Theorem (Continuity and boundedness). A linear lunctional I with domain CZ/;(f) in a normed space is continuous il and only il I is bounded. I Chapter.4.1-4.5 Brian P. Rynne and Martin A. Youngson, Linear Functional Analysis Chapter.5.1-5.3 Marián Fabian,Petr Habala,Petr Hájek, Vicente Montesinos,Václav Zizler, Banach Space Theory, The Basis for Linear and Nonlinear Analysis Section 2.1-2.2 VI. This Cauchy sequence does not converge. The proof is the same as in 1.5-9, with the metric in 1.5-9 replaced by the present metric. For a general interval [a, b] we can construct a similar'" Cauchy sequence which does not converge in X. The space X can be completed by Theorem 1.6-2. The completion is denoted by L 2[a, b]. This is a Banach space. In fact, the norm on X and the operations of vector space can be extended to the completion of X, as we shall see from Theorem 2.3-2 in the next section. More generally, for any fixed real number p ~ 1, the Banach space a( Cx)(f) + (3( Cy)(f). C is also called the canonical embedding of X into X**. To understand and motivate this term, we first explain the concept of "isomorphism," which is of general interest. In our work we are concerned with various spaces. Common to all of them is that they consist