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Chapter 1 Introduction Chapter 1 Introduction Functional analysis can be seen as a natural extension of the real analysis to more general spaces. As an example we can think at the Heine - Borel theorem (closed and bounded is equivalent with compact) which will be now extended to any finite dimensional space. Moreover, it will be shown that the theorem is not true in an infinite dimensional space. Functional analysis is furnishing plenty of techniques of proofs, the proof be- ing sometimes more important as the result itself. A special attention should be given to the proofs presented in this lecture (including the exercises). Another benefit in learning functional analysis is that one reaches a higher level of abstractization when dealing with mathematical objects and understands that many times is easier to prove for a general case as for a particular case. The latter is because when you deal with an explicitly described mathematical object (e.g. a space) one does not know which properties are now relevant for the specific question to be answered (in other words one has a problem of too much informa- tion). It is often much easier to answer the question when you think about the object as part of a class, means trying to identify properties which are general, valid for many objects of a certain type. As an example: imagine you have to show that the space of continuous functions on the interval [0; 1], denoted in this script C[0; 1] can not have an infinite but countable Hamel basis. The proof is in one line if one just know that not any Banach space can have such a basis (see Section 2.7 for details). There are also many practical applications of functional analysis. The most direct ones are connected with solving partial differential equations (PDEs), with a special focus on numeric. Solving PDEs is extremely important for real world applications, because the mathematical models for such applications are consist- 1 CHAPTER 1. INTRODUCTION 2 ing in the majority of the cases by PDEs. When modelling real worlds problems, the PDEs have often e.g. discontinuous coefficients (due to the heterogeneity of the medium) or are degenerate (e.g. from parabolic to elliptic in the case of variable saturated flow in porous media [13, 14]) and do not admit a classical solution. Functional analysis provides the proper mathematical framework (e.g. spaces, norms, techniques) for defining weak solutions for PDEs and solving them numerically. Chapter 2 Banach spaces 2.1 Normed (linear) spaces Let V be a (linear) vector space. Definition 2.1.1 A functional k · k : V ! R is a norm on V over R if it satisfies the following: i) kλxk = jλjkxk, for all x 2 V; λ 2 R: ii) kx + yk ≤ kxk + kyk; for all x; y 2 V: (the triangle inequality) iii) kxk = 0 ) x = 0: If k · k satisfies only i)-ii) it is called a seminorm. Remark 2.1.1 If k · k is a norm on a vector space V , it follows from i) and ii) that kxk ≥ 0 8 x 2 V . Often is this property included in the definition of the norm, but it is not necessary. Definition 2.1.2 Two norms k · k1; k · k2 are equivalent on the space V if there exists m; M > 0 s.t. there holds mkxk1 ≤ kxk2 ≤ Mkxk1 8 x 2 V: We mention that there exists also the notion of quasi-norm (see e.g. Yosida, pg. 30). There are nevertheless a few different definitions in the literature. Let V a vector space and k · kV a norm on it. The norm induces a metric on V defined by d(x; y) = kx − ykV . Using this metric we can define a topology, which is called the topology given by the norm k · kV . A topological vector space 3 CHAPTER 2. BANACH SPACES 4 X is a space which has both an algeberaic structure (vector space) and a topology on it, and the addition and multiplication are continuous (as functions from Γ × X to X, where Γ is the field of scalars). Definition 2.1.3 A normed space (V; k·kV ) is a topological vector space V , with the topology given by the norm k · kV . A very important inequality which holds in any normed space (V; k · kV ) is the following jkxk − kykj ≤ kx − yk; (2.1) for any x; y 2 V . The proof is straightforward. Examples of normed spaces In this section we present some important normed spaces. We define for 1 ≤ p < 1 the p-norm on Rn by n !1=p X p kxkp := jxij (2.2) i=1 n Proposition 2.1.1 For all 1 ≤ p < 1, k · kp is indeed a norm on R . Moreover, n (R ; k · kp) is a Banach space. For p = 1 we define n kxk1 := max jxij (2.3) i=1 n n Proposition 2.1.2 k · k1 is a norm on R . Moreover, (R ; k · k1) is a Banach space. All the above spaces are finite dimensional. As examples of infinite dimen- sional Banach spaces we consider C[0; 1] := ff : [0; 1] ! Rjf continuous g with the norm kfk1 = max jf(x)j: (2.4) x2[0;1] and the Sobolev spaces p L (Ω) := ff :Ω ! Rjf measurable and kfkp < 1g; where Ω is an open set in Rd, (d 2 N; d ≥ 1) and the norm Z 1=p p kfkp := jf(x)j dx ; (2.5) Ω CHAPTER 2. BANACH SPACES 5 for 1 ≤ p < 1 and kfk1 = (ess) sup jf(x)j; (2.6) x2[0;1] for p = 1. We remind that the essential supremum of a measurable function f() on a domain Ω is defined as the lowest number α 2 R, for which f(x) ≤ α almost everywhere in Ω. Definition 2.1.4 A subspace E of V is closed iff V n E is open, or, equivalently, for any sequence fxngn2N ⊂ E, if xn ! x then x 2 E. The closure of a subspace (set) E is denoted by E and it is the smallest set which contains E and is closed. E := fx 2 V j9 fxngn2N ⊂ E with xn ! xg: Definition 2.1.5 A topological vector space is called Hausdorff if for any two element x; y 2 X it exists two open sets Vx and Vy such that x 2 Vx, y 2 Vy and Vx \ Vy = ;. A topological vector space is called separable if it has a countable, dense set. It is easy to check that any metric space (therefore also every normed space) is Hausdorff. 2.1.1 Exercises Exercise 2.1.1 Prove that in any normed space the conditions kxk = 1 and kx − y yk < < 1 imply that kx − k < 2. kyk Exercise 2.1.2 Prove that in a normed space, if kx + yk = kxk + kyk, then kαx + βyk = kαxk + kβyk for all non-negative real numbers α; β. Exercise 2.1.3 Let X be a vector space and k · k1; k · k2 two norms on it. i) Let kxkmax := max(kxk1; kxk2). Show that the function k · kmax defined by kxkmax := max(kxk1; kxk2) for all x 2 X is a norm on X as well. Does the same hold also for kxkmin := min(kxk1; kxk2)? ii) Let α; β 2 R be two positive numbers. Show that the application k · k defined by kxk := αkxk1 + βkxk2 for all x 2 X is a norm on X. iii) Let p ≥ 1 and jj · jjp be defined by p p 1=p jjxjjp = (kxk1 + kxk2) ; for all x 2 X, is a norm on X. CHAPTER 2. BANACH SPACES 6 Exercise 2.1.4 We define for 1 ≤ p < 1 the p-norm on Rn by n !1=p X p kxkp := jxij : i=1 Show that k · kp is indeed a norm. Exercise 2.1.5 Let (X; k · k) be a normed space, fαngn ⊂ R, fxngn; fyngn ⊂ X and fαgn ⊂ R, x; y 2 X. Show that the following affirmations hold true: i) If xn ! x strongly (i.e. in norm) and αn ! α, then there holds limn!1(αnxn) = αx. ii) If xn ! x and yn ! y strongly, then xn + yn ! x + y strongly. iii) If xn ! x and xn ! y strongly then x = y (the limit of a sequence is unique). iv) If xn ! x strongly, then kxn − yk ! kx − yk in R. Especially, for y = 0 it results that kxnk ! kxk in R. 2.2 Completeness, Banach spaces Let (X; k · k) be a normed space. Definition 2.2.1 A sequence fxngn ⊂ X is called Cauchy if 8 > 09N 2 N such that kxn − xmk ≤ , 8n; m ≤ N: Remark 2.2.1 Any convergent sequence is Cauchy, but not any Cauchy sequence is convergent. A counter-example is the space of all sequences in R with a finite number of nonzero elements. Definition 2.2.2 A topological vector space V is called complete if every Cauchy sequence is convergent to an element of V. A normed vector space which is com- plete is called a Banach space. Remark 2.2.2 Not any normed space is a Banach space. A counter-example is the space of all sequences in R with a finite number of nonzero elements, with the infinite norm. CHAPTER 2. BANACH SPACES 7 Theorem 2.2.1 The space C[0; 1] := ff : [0; 1] ! Rjf continuous g with the norm kfk1 = maxx2[0;1] jf(x)j is a Banach space.
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