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

Product Information

Furthermore, the series in (5) represents [d(xm, x)]P, so that (5) implies that Xm x. Since (xm) was an arbitrary Cauchy sequence in lV, this proves completeness of IV, where 1 ~ P < +00. • 1.5-5 Completeness of C[ a, b]. The function space C[ a, b] is complete; here [a, b] is any given closed interval on R. (Cf. 1.1-7.) Proof. Let (Xm) be any Cauchy sequence in C[a, b]. Then, given any e > 0, there is an N such that for all m, n> N we have Sequences were available to us even in a general metric space. In a narmed space we may go an important step further and use series as fu~~.

say, ~;m) ~J as m _ 00. Using these limits, we define (~}, ~2' . . . ) and show that x E lV and Xm x. From (3) we have for all m, n> N

Introductory Functional Analysis with Applications

m>N). This shows that x is the limit of (xm) and proves completeness of R n bccause (xm) was an arbitrary Cauchy sequence. Completeness of C n follows from Theorem 1.4-4 by the same method of proof. 1.5-2 Completeness of l"". We see that an open ball of radius r is the set of all points in X whose distance from the center of the ball is less than r. Furthermore, the definition immediately implies that (2) complete normed space (complete in the metric defined by the norm; see (1), below). Here a nonn on a (real or complex) vector space X is a real-valued function on X whose value at an x E X is denoted by

is the completion of the normed space which consists of all continuous real-valued functions on [a, b], as before, and the norm defined by (8) Topics In Complex Function Theory. Volume III ―Abelian Functions & Modular Functions of Several Variables Space e[a, b]. Another practically important functional on C[a, bJ is obtained if we choose a fixed to E J = [a, bJ and set X E 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 319This shows that Xm - x = (~im) - ~j) E lV. Since Xm E IV, it follows by means of the Minkowski inequality (12), Sec. 1.2, that Proof. Let (xn) be any Cauchy sequence in the space [P, where (~im), ~~m\ •• '). Then for every E > 0 there is an N such that for all 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 Space 12. We can obtain a linear functional f on the Hilbert space [2 (cf. 1.2-3) by choosing a fixed a = (aj) E [2 and setting 00

because x EM was arbitrary. We prove that M is bounded. If M were unbounded, it would contain an unbounded sequence (Yn) such that d(Yn, b» n, where b is any fixed element. This sequence could not have a convergent subsequence since a convergent subsequence must be bounded, by Lemma 1.4-2 . •

show that the Cauchy sequences in (X, d 1 ) and (X, dz) are the same. 9. Using Prob. 8, show that the metric spaces in Probs. 13 to 15, Sec. 1.2, have the same Cauchy sequences. Proof. Let y = (TJl. TJz, TJ3, ••• ) be a sequence of zeros and ones. Then y E I"". With Y we associate the real number y whose binary representation is

Observe the notation; we write Tx instead of T(x); this simplification is standard in functional analysis. Furthermore, for the remainder The concept of convergence of a series can be used to define a "basis" as follows. If a normed space X contains a sequence (en) with the property that for every X E X there is a unique sequence of scalars (un) such that (as

Discover

Tzafriri Valery Serov: Fourier Series, Fourier Transform and Their Applications to Mathematical Physics Obviously, (Xn) cannot have a convergent subsequence. This contradicts the compactness of M. Hence our assumption dim X = 00 is false, and dim X < 00. • This theorem has various applications. We shall use it in Chap. 8 as a basic tool in connection with so-called compact operators. KREYSZIG 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. Currently available in the Emil Artin 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. Can every incomplete normed space be completed? As a metric space certainly by 1.6-2. But what about extending the operations of a vector space and the norm to the completion? We shall see in the next section that the extension is indeed possible.