Fig. 4. Euclidean metric on the plane 4 We do not write x = (XI> X2) since x" X2, sequences (starting in Sec. 1.4).

hump method Terry J. Morrison, Functional Analysis: An Introduction to Banach Space Theory, 2001 John Wiley & Sons, p.77. Albrecht Pietsch, History of Banach Spaces and Linear Operators,2007 Birkhauser Boston, p.41 S okal ,Alan (2011),"A really simple elementary proof of the uniform boundedness theorem", Amer. Math. Monthly, 118 : 450–452, arXiv : 1005.1585 , doi : 10.4169/amer.math.monthly.118.05.450 . For every x E M there is a sequence (x..) in M such that x; cf. 1.4-6(a). Since M is compact, x E M Hence M is closed where n is any positive integer and the '1//s are rational. M is countable. We show that M is dense in IP. Let x = (g) E lP be arbitrary. Then for every 8> there is an n (depending on 8) such that 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.Problems 1. (Subsequence) If a sequence (x..) in a metric space X is convergent and has limit x, show that every subsequence (x...) of (xn) is convergent and has the same limit x. 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 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.) The book is suitable for a one-semester course meeting five hours per week or for a two-semester course meeting three hours per week. The book can also be utilized for shorter courses. In fact, chapters can be omitted without destroying the continuity or making the rest of the book a torso 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

From this theorem and Theorem 1.4-7 we have 2.4-3 Theorem (Closedness). Every finite dimensional subspace Y of a normel1 space X is closed in X. We shall need this theorem at several occasions in our further work. Note that infinite dimensional subspaces need not be close'd. Example. Let X=C[O,1] and Y=span(xo,xl,···), where Xj(t)=t i , so that Y is the set of all polynomials. Y is not closed in X. (Why?) 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. Definition (Bounded linear functional). A bounded linear lunctional I is a bounded linear operator (ef. Def. 2.7-1) with range in the scalar field of the normed space X in which the domain 9J(f) lies. Thus there exists a real number c such that for all x E 9J(f),and is called the metric induced by the norm. The normed space just defined is denoted by (X, II . II) or simply by X. • The defining properties (Nl) to (N4) of a norm are suggested and motivated by the length Ixl of a vector x in elementary vector algebra, so that in this case we can write Ilxll = Ixl. In fact, (Nl) and (N2) state that all vectors have positive lengths except the zero vector which has length zero. (N3) means that when a vector is multiplied by a scalar, its length is multiplied by the absolute value of the scalar. (N4) is illustrated in Fig. 15. It means that the length of one side of a triangle cannot exceed the sum of the lengths of the two other sides. It is not difficult to conclude from (Nl) to (N4) that (1) does define a metric. Hence normed spaces and Banach spaces are metric spaces. For example, IP in 2.2-3 has a Schauder basis, namely (en), where en = (8,.j), that is, en is the sequence whose nth term is 1 and all other This theorem is of considerable practical importance. For instance, it implies that convergence or divergence of a sequence in a finite dimensional vector space does not depend on the particular choice of a norm on that space.

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), Quotient space, codimension) Let Y be a subspace of a vector space X. The coset of an element X E X with respect to Y is denoted by X + Y and is defined to be the set (see Fig. 12) x+Y={vlv=X+y,YEY}.

Develop

Marián Fabian,Petr Habala,Petr Hájek, Vicente Montesinos,Václav Zizler, Banach Space Theory, The Basis for Linear and Nonlinear Analysis Examples of Schauder Basis pp. 185-187 III.

About Author :- Erwin O. Kreyszig (1922–2008) was a German Canadian applied mathematician and the Professor of Mathematics at Carleton University in Ottawa, Ontario, Canada. He was a pioneer in the field of applied mathematics: non-wave replicating linear systems. He was also a distinguished author, having written the textbook Advanced Engineering Mathematics, the leading textbook for civil, mechanical, electrical, and chemical engineering undergraduate engineering mathematics. This was my textbook for a graduate course in functional analysis, and it is called "classic" by many professors. Don't be fooled by the title of the book: "Introductory" simply means the author assumes you have not seen the subject before, and it is by no means an easy subject. However, the exposition is extremely clear. Kreyszig saved me on numerous occasions as my companion on a treacherous journey through graduate functional analysis. 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.

My Book Notes

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