Kreyszig received his Ph.D. degree in 1949 at the University of Darmstadt under the supervision of Alwin Walther. Definition (Linear functional). A linear lunctional I is a linear operator with domain in a vector space X and range in the scalar field K of X; thus, We see directly from the definition that every neighborhood of Xo contains Xo; in other words, Xo is a point of each of its neighborhoods. And if N is a neighborhood of Xo and N eM, then M is also a neighborhood of Xo. We call Xo an interior point of a set Me X if M is a neighborhood of Xo. The interior of M is the set of all interior points of M and may be denoted by ~ or Int (M), but there is no generally accepted notation. Int (M) is open and is the largest open set contained in M. It is not difficult to show that the collection of all open subsets of X, call it fT, has the follpwing properties: (Tl) Show that for an infinite subset M in Ihe space s (d. 2.2-8) to be compact, it is necessary that there arc numhers 1'1,1'2, ... such that for all X=(~k(X»EM we have l~k(X)I~1'k. (II can he shown that the condition is also sufficient for the compactness of M.l S. (Local compactness) A metric space X is said to be locally compact if every point of X has a compact neighborhood. Show that Rand C and, more generally, R" and C" are locally compact. 6. Show that a compact metric space X is locally compact. 7. If dim Y < 00 in Riesz's lemma 2.5-4, show that one can even choose 8 = 1. 8. In Prob. 7, Sec. 2.4, show directly (without using 2.4-5) that there is an a> 0 such that a Ilxlb:;;;; Ilxll. (Use 2.5-7.) 9. If X is a compact metric space and Me X is closed, show that M is compact. 10. Let X and Y be metric spaces, X compact, and T: X ~ Y bijective and continuous. Show that T is a homeomorphism (d. Prob. 5, Sec. We choose a fixed j. From (4) we see that (~?\ ~F), ... ) is a Cauchy sequence of numbers. It converges since Rand C are complete (cf.

Finite Dimensional Normed Spaces and Suhspaces Are finite dimensional normed spaces simpler than infinite dimensional ones? In what respect? These questions are rather natural. They are important since finite dimensional spaces and subspaces play a role in various considerations (for instance, in approximation theory and spectral theory). Quite a number of interesting things can be said in this connection. Hence it is worthwhile to collect some relevant facts, for their own sake and as tools for our further work. This is our program in this section and the next one. A source for results of the desired type is the following lemma. Very roughly speaking it states that in the case of linear independence of vectors we cannot find a linear combination that involves large scalars but represents a small vector. 2.4-1 Lemma (Linear combinations). Let {Xl, . . . ,x,.} be a linearly independent set of vectors in a normed space X (of any dimension). Then there is a number c > 0 such that for every choice of scalars al> .•. , an we have (c > 0).Before we go on, we mention another unusual property of balls in a metric space. Whereas in R3 the closure B(xo; r) of an open ball B(xo; r) is the closed ball B(xo; r), this may not hold in a general metric space. We invite the reader to illustrate this with an example. Using the concept of the closure, let us give a definition which will be of particular importance in our further work: Theorem (Completion). Let X = (X, 11·11) be a normed space. Then there is a Banach space X and an isometry A from X onto a subspace W of X which is dense in X. The space X is unique, except for isometries. Proof. Theorem 1.6-2 implies the existence of a complete metric space X = d) and an isometry A: X W = A (X), where W is dense in X and X is unique, except for isometries. (We write A, not T as in 1.6-2, to free the letter T for later applications of the theorem in Sec. 8.2) Consequently, to prove the present theorem, we must make X into a vector space and then introduce on X a suitable norm. To define on X the two algebraic operations of a vector space, we consider any X, y E X and any representatives (x..) E X and (Yn) E y. Remember that x and yare equivalence classes of Cauchy sequences in X. We set Zn = Xn + Yn' Then (zn) is Cauchy in X since The problems given in the book help the students further understand the principles explained in the book. These exercises provide the student with practice at applying the concepts learnt to solve problems. The book also provides illustrations of practical applications of the theorems covered. It touches on the practical uses of these principles, not just in mathematics, but also in other branches of study like physics. lies in a vector space over the same field, 5 Some familiarity with the concept of a mapping and simple related concepts is assumed, but a review is included in A1.2; d. Appendix 1. Banach Fixed Point Theorem 299 5.2 Application of Banach's Theorem to Linear Equations 5.3 Applications of Banach's Theorem to Differential Equations 314 5.4 Application of Banach's Theorem to Integral Equations 319

because on the left we have the remainder of a converging series. Since the rationals are dense in R, for each ~j there is a rational '1/j close to it. Hence we can find ayE M satisfying Compactness and Finite Dimension A few other basic properties of finite dimensional normed spaces and subspaces are related to the concept of compactness. The latter is defined as follows. 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 516 Show that a discrete metric space X (cf. 1.1-8) consisting of infinitely many points is not compact. 3. Give examples of compact and noncompact curves in the plane R2. The elements X" X2 generate a two dimensional proper closed subspace X 2 of X. By Riesz's lemma there is an X3 of norm 1 such that for all X E X 2 we have 1

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C n is sometimes called complex Euclidean n-space.) 1.1-6 Sequence space l"'. This example and the next one give a first impression of how surprisingly general the concept of a metric spa 0, we define 6 three types of sets: (a) Functional analysis is a form of mathematical analysis that originated from classical analysis. It is finding increasing applications in mathematics and in natural sciences. This book, Introductory Functional Analysis With Applications, is intended to introduce and explain the subject to college students. So, it focuses on the fundamental principles, theory and also their practical applications. Proof. A countable dense subset of C is the set of all complex numbers whose real and imaginary parts are both rational._ Definition (Equivalent norms). A norm 11·11 on a vector space X is said to be equivalent to a norm I . 110 on X if there are positive numbers a and b such that for all x

Examples of Incomplete Metric Spaces 1.5-7 Space Q. This is the set of all rational numbers with the usual metric given by d(x, y)=lx-yl, where x, YEQ, and is called the rational line. Q is not complete. (Proof?) 1.5-8 Polynomials. Let X be the set of all polynomials considered as functions of t on some finite closed interval J = [a, b] and define a I is linear. We prove that I is bounded and has norm IIIII = b - a. In fact, writing J = [a, b] and remembering the norm on C[a, b], we obtain where x = ({;j) and y = ('fjj)] and (xm) is Cauchy, for any 8> 0 there is an N such that for all m, n > N, d(x m, xn) = sup I{;~m) j Discrete metric space. A discrete metric space X is separable if and only if X is countable. (Cf. 1.1-8.) Proof. The kind of metric implies that no proper subset of X can be dense in X. Hence the only dense set in X is X itself, and the statement follows.Compact sets are important since they are "well-behaved": they have several basic properties similar to those of finite sets and not shared by noncompact sets. In connection with continuous mappings a fundamental property is that compact sets have compact images, as follows. 2.5-6 Theorem (Continuous mapping). Let X and Y be metric spaces and T: X Ya continuous mapping (cf. 1.3-3). Then the image of a compact subset M of X under T is compact. Proof. By the definition of compactness it suffices to show that every sequence (Yn) in the image T(M) c Y contains a subsequence which converges in T(M). Since Yn E T(M), we have Yn = Tx,. for some Xn EM Since M is compact, (xn) contains a subsequence (xn,J which converges in M. The image of (x n.) is a subsequence of (Yn) which converges in~(M) by 1.4-8 because T is continuous. Hence T(M) is compact. I From this theorem we conclude that the following property, well-known from calculus for continuous functions, carries over to metric spaces. 2.5-7 Corollary (Maximum and minimum). A continuous mapping T of a compact subset M of a metric space X into R assumes a maximum and a minimum at some points of M. Proof. T(M) c R is compact by Theorem 2.5-6 and closed and bounded by Lemma 2.5-2 [applied to T(M)], so that inf T(M)E T(M), sup T(M) E T(M), and the inverse images of these two points consist of points of M at which Tx is minimum or maximum, respectively. I Infinite series can now be defined in a way similar to that in calculus. In fact, if (Xk) is a sequence in a normed space X, we can associate with (Xk) the sequence (sn) of partial sums sn Theorem (Compactness). In a finite dimensional normed space X, any subset M c X is compact if and only if M is closed and bounded. 4 More precisely, sequentially compact; this is the most important kind of compactness in analysis. We mention that there are two other kinds of compactness, but for metric spaces the three concepts become identical, so that the distinction does not matter in our work. (The interested reader will find some further remarks in A 1.5. Appendix 1.)

We shall present completeness proofs for some metric spaces which occur quite frequently in theoretical and practical investigations. Unbounded Linear Operators and their Hilbert-Adjoint Operators 524 10.~ Hilbert-Adjoint Operators, Symmetric and Self-Adjoint Linear Operators 530 10.3 Closed Linear Operators and Cldsures 535 10.4 Spectral Properties of Self-Adjoint Linear Operators 541 10.5 Spectral Representation of Unitary Operators 546 10.6 Spectral Representation of Self-Adjoint Linear Operators 556 10.7 Multiplication Operator and Differentiation Operator 562 Erwin Kreyszig was a professor and mathematician. Introduction to Differential Geometry and Riemannian Geometry, and Advanced Engineering Mathematics are also written by this author. We now use the facts that the set of points in the interval [0,1] is uncountable, each yE [0, 1] has a binary representation, and different fs have different binary representations. Hence there are uncountably many sequences of zeros and ones. The metric on I"" shows that any two of them which are not equal must be of distance 1 apart. If we let as the reader may readily prove (cf. Prob. 3). Formula (2) implies an important property of the norm: The norm is continuous, that is, x ~ of (X, 11·11) into R. (Cf. 1.3-3.)

Introductory Functional Analysis with Applications

Theorem (Uniform convergence). Convergence Xm ~ x in the space C[a, b] is uniform convergence, that is, (Xm) converges uniformly on [a, b] to x. Hence the metric on C[a, b] describes uniform convergence on [a, b] and, for this reason, is sometimes called the uniform metric. To gain a good understanding of completeness and related concepts, let us finally look at some