DOCSLIB.ORG

Lecture Notes Functional Analysis WS 2012/2013

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Lecture Notes Functional Analysis WS 2012/2013 February 13, 2013 2 Contents I Normed vector spaces, Banach spaces and metric spaces5 1 Normed vector spaces and Banach spaces . .5 2 Basics of metric spaces . .9 3 Compactness in metric space . 19 4 The sequence spaces lp(N); 1 ≤ p ≤ 1 ............... 23 5 Hahn-Banach type theorems . 27 5.1 Some preparations . 27 5.2 The analytic form of Hahn-Banach: extension of linear functionals . 30 5.3 Geometric form of Hahn-Banach . 34 6 The Baire Category theorem and its applications . 39 6.1 The Baire Category theorem . 39 6.2 Application I: The set of discontinuities of a limit of con- tinuous functions . 41 6.3 Application II: Continuous but nowhere differentiable func- tions . 43 6.4 Application III: The uniform boundedness principle . 44 6.5 Application IV: The Open Mapping and the Closed Graph theorems . 46 7 Weak Topologies. Reflexive Spaces. Separable Spaces. Uniform Convexity . 49 7.1 The coarsest topology for which a collection of maps be- comes continuous . 49 7.2 The weakest topology σ(E; E∗)............... 51 7.3 Weak topology and convex sets . 55 7.4 The weak∗ topology σ(E∗;E)................ 57 7.5 Reflexive spaces . 60 7.6 Separable spaces . 64 7.7 Uniformly convex spaces . 68 8 Lp spaces . 71 8.1 Some results from integration everyone must know . 71 8.2 Definition and some properties of Lp spaces . 73 8.3 Reflexivity, Separability. The Dual of Lp .......... 77 9 Hilbert spaces . 87 9.1 Some elementary properties . 87 9.2 The dual space of a Hilbert space . 94 9.3 The Theorems of Stampacchia and Lax-Milgram . 96 3 4 CONTENTS Chapter I Normed vector spaces, Banach spaces and metric spaces 1 Normed vector spaces and Banach spaces In the following let X be a linear space (vector space) over the field F 2 fR; Cg. Definition 1.1. A seminorm on X is a map p : X ! R+ = [0; 1) s.t. (a) p(αx) = jαjp(x) 8α 2 F; 8x 2 X (homogeneity). (b) p(x + y) ≤ p(x) + p(y) 8x; y 2 X (triangle inequality). If, in addition, one has (c) p(x) = 0 ) x = 0 then p is called a norm. Usually one writes p(x) = kxk, p = k · k. The pair (X; k · k) is called a normed (vector) space. Remark 1.2. • If k · k is a seminorm on X then kxk − kyk ≤ kx − yk 8x; y 2 X (reverse triangle inequality): Proof. kxk = kx − y + yk ≤ kx − yk + kyk ) kxk − kyk ≤ kx − yk Now swap x&y: kxk − kyk ≤ ky − xk = k(−1)(x − y)k = kx − yk. Hence kxk − kyk = max(kxk − kyk; kyk − kxk) ≤ kx − yk: 5 6 CHAPTER I. NVS, BS AND MS •k ~0k = k0:~0k = j0j:k~0k = 0. Interpret kx − yk as distance between x and y. Definition 1.3. Let (xn)n2N = (xn)n be a sequence in a normed vector space X ((X; k · k)). Then (xn)n converges to a limit x 2 X if 8" > 09N" 2 N s.t. 8n ≥ N" it holds kxn − xk < " (or kxn − xk ≤ "). One writes xn ! x or limn!1 xn = x. (xn)n is a Cauchy sequence if 8" > 09N" 2 N s.t. 8n; m ≥ N" it holds kxn − xmk < " (or kxn − xmk ≤ "). (X; k · k) is complete if every Cauchy sequence converges. A complete normed space (X; k · k) is called a Banach space. Remark 1.4. Let X be a normed vector space, (xn)n a sequence in X. (a) If xn ! x in X, then (xn)n is a Cauchy sequence. " Proof. Given " > 09N" : 8n ≥ N" : kx − xnk < 2 . Hence for n; m ≥ N" we have kxn − xmk = kxn − x + x − xmk ≤ kxn − xk + kx − xmk < ": (b) Limits are unique! If xn ! x in X and xn ! y in X, then x = y. Proof. kx − yk = kx − xn + xn − yk ≤ kxn − xk + kxn − yk ! 0 + 0 = 0 as n ! 1: (c) If (xn)n converges or is Cauchy, then it is bounded, i.e. sup kxnk < 1: n2N Proof. Take " = 1. Then there exists N 2 N s.t. 8n; m ≥ N : kxn − xmk < 1. In particular, 8n ≥ N : kxn − xN k < 1: ) kxnk = kxn − xN + xN k ≤ kxn − xN k + kxN k < 1 + kxN k (8n 2 N) ) 8n 2 N : kxnk ≤ max(kx1k; kx2k;:::; kxN k; 1 + kxN k) < 1: 1. NORMED VECTOR SPACES AND BANACH SPACES 7 Let X be a normed vector space, S 6= ; a set. For functions f; g : S ! X, α; β 2 F define ( S ! X f + g : s 7! (f + g)(s) = f(s) + g(s) ( S ! X αf : s 7! (αf)(s) = αf(s) So the set of functions from S to X is a normed space itself! In case X = R, we write f ≥ α (or f > α) if f(s) ≥ α for all s 2 S (f(s) > α for all s 2 S). Similarly one defines α ≤ f ≤ β, f ≤ g, etc. Example 1.5. (a) X = Fd; d 2 N is a Banach space (or short B-space) with respect to (w.r.t) the norms n 1 X p p jxjp := jxjj ; 1 ≤ p < 1; j=1 jxj1 := max jxjj: j=1;:::;d d Here x = (x1; x2; : : : ; xd) 2 F . (b) Let Ω 6= ;, L1(Ω) = L1(Ω; R) = the set of all real-valued functions on Ω which are bounded, i.e. 1 f 2 L (Ω) then 9Mf < 1 : jf(!)j ≤ Mf for all ! 2 Ω: 1 1 Norm on L (Ω): for f 2 L (Ω) : kfk1 = sup!2Ω jf(!)j (check that this is a norm!). 1 Claim: (L (Ω); k · k1) is a Banach space. Proof. Normed vector space is clear. 1 Take (fn) a Cauchy sequence in L (Ω) w.r.t. k · k1. We have: 8" > 09N : 8n; m ≥ N : kfn − fmk1 < ". Fix ! 2 Ω, then fn(!) is a Cauchy sequence in R since jfn(!) − fm(!)j ≤ sup jfn(!) − fm(!)j = kfn − fmk1 < " 8n; m ≥ N: !2Ω Since R is complete, f(!) := limn!1 fn(!) exists (this f is the candidate for the limit). We have jf(!)j ≤ jf(!) − fn(!)j + jfn(!)j = lim jfm(!) − fn(!)j + jfn(!)j ≤ " + jfn(!)j m!1 | {z } <1 ) sup!2Ωjf(!)j ≤ 1; i.e., f 2 L1(Ω). Take " > 0. Then jfn(!) − f(!)j = lim jfn(!) − fm(!)j ≤ " if n ≥ N m!1 | {z } ≤" if n;m≥N 8 CHAPTER I. NVS, BS AND MS ) 8n ≥ N : kfn − fk1 ≤ "; i.e., fn ! f w.r.t. k · k1: 1 R (c) X = C([0; 1]); kfk1 := jf(t)jdt is a norm, C([0; 1]); k·k1 is not complete. 0 Proof. kfk1 ≥ 0 8f 2 C([0; 1]): 1 Z kf + gk1 = jf(t) + g(t)j dt ≤ kfk1 + kgk1 | {z } 0 ≤|f(t)j+jg(t)j 1 Z kαfk1 = jαf(t)jdt = jαjkfk1 0 So k · k is a seminorm. If f 6≡ 0 and f is continuous, we see that there exist an interval I ⊂ [0; 1], δ > 0 such that jf(t)j ≥ δ 8t 2 I. 1 Z Z ) kfk1 = jf(t)jdt ≥ jf(t)j dt ≥ δ:lehgth of I > 0: | {z } 0 I ≥δ So k · k1 is a seminorm. Now take a special sequence 8 1 1 >0; if 0 ≤ t ≤ 2 − n < n 1 1 1 fn(t) := nt − 2 + 1; if 2 − n < t < 2 (n ≥ 3) :> 1 1; if t ≥ 2 For m ≥ n ≥ 3: 1 n Z 1 kf − f k = jf (t) − f (t)jdt ≤ ! 0 as n ! 1; n m 1 n m n 1 1 2 − n so (fn) is a Cauchy sequence. 1 1 1 Assume that fn ! f 2 C([0; 1]). Fix α 2 0; 2 ; n : 2 − n ≥ α α α Z Z 0 ≤ jf(t)jdt = jfn(t) − f(t)jdt 0 0 1 Z ≤ jfn(t) − f(t)jdt = kfn − fk1 ! 0: 0 1 Hence f(t) = 0 for all 0 ≤ t ≤ α, all 0 ≤ α < 2 1 f(t) = 0 for all0 ≤ t < : 2 2. BASICS OF METRIC SPACES 9 On the other hand 1 1 Z Z 0 ≤ jf(t) − 1jdt = jf(t) − fn(t)jdt ≤ kf − fnk1 ! 0 as n ! 1 1 1 2 2 1 Since f is continuous on [0; 1], it follows that f(t) = 1 for 2 ≤ t ≤ 1. 1 So f cannot be continuous at t = 2 . A contradiction. 2 Basics of metric spaces Definition 2.1. Given a set M 6= ;, a metric (or distance) d on M is a function d : M × M ! R such that (a) d(x; y) ≥ 0 8x; y 2 M and d(x; y) = 0 () x = y. (b) d(x; y) = d(y; x) 8x; y 2 M (symmetry). (c) d(x; y) ≤ d(x; z) + d(z; y) 8x; y; z 2 M (triangle inequality). The pair (M; d) is called a metric space. We often simply write M if it is clear what d is. A sequence (xn)n in a metric space (M; d) converges to x 2 M if 8" > 09N" 2 N8n ≥ N" : d(x; xn) < " (or ≤ "). One writes lim xn = x or xn ! x. One always has d(x; z) − d(z; y) ≤ d(x; y) Hint for the proof: d(x; z) ≤ d(x; y) + d(y; z) and think and use symmetry. Example 2.2. • R with d(x; y) = jx − yj; • Any normed vector space (X; k · k) with d(x; y) = kx − yk; d 1 d d P 2 2 • Eucledian space R (or C ) with d2(x; y) = jxj −yjj or dp(x; y) = j=1 d 1 P p p jxj − yjj , or d1(x; y) = maxj=1;:::d jxj − yjj.
Recommended publications
  • On Quasi Norm Attaining Operators Between Banach Spaces

    On Quasi Norm Attaining Operators Between Banach Spaces

    ON QUASI NORM ATTAINING OPERATORS BETWEEN BANACH SPACES GEUNSU CHOI, YUN SUNG CHOI, MINGU JUNG, AND MIGUEL MART´IN Abstract. We provide a characterization of the Radon-Nikod´ymproperty in terms of the denseness of bounded linear operators which attain their norm in a weak sense, which complement the one given by Bourgain and Huff in the 1970's. To this end, we introduce the following notion: an operator T : X ÝÑ Y between the Banach spaces X and Y is quasi norm attaining if there is a sequence pxnq of norm one elements in X such that pT xnq converges to some u P Y with }u}“}T }. Norm attaining operators in the usual (or strong) sense (i.e. operators for which there is a point in the unit ball where the norm of its image equals the norm of the operator) and also compact operators satisfy this definition. We prove that strong Radon-Nikod´ymoperators can be approximated by quasi norm attaining operators, a result which does not hold for norm attaining operators in the strong sense. This shows that this new notion of quasi norm attainment allows to characterize the Radon-Nikod´ymproperty in terms of denseness of quasi norm attaining operators for both domain and range spaces, completing thus a characterization by Bourgain and Huff in terms of norm attaining operators which is only valid for domain spaces and it is actually false for range spaces (due to a celebrated example by Gowers of 1990). A number of other related results are also included in the paper: we give some positive results on the denseness of norm attaining Lipschitz maps, norm attaining multilinear maps and norm attaining polynomials, characterize both finite dimensionality and reflexivity in terms of quasi norm attaining operators, discuss conditions to obtain that quasi norm attaining operators are actually norm attaining, study the relationship with the norm attainment of the adjoint operator and, finally, present some stability results.
  • 209 a Note on Closed Graph Theorems

    209 a Note on Closed Graph Theorems

    Acta Math. Univ. Comenianae 209 Vol. LXXV, 2(2006), pp. 209–218 A NOTE ON CLOSED GRAPH THEOREMS F. GACH Abstract. We give a common generalisation of the closed graph theorems of De Wilde and of Popa. 1. Introduction In the theory of locally convex spaces, M. De Wilde’s notion of webs is the abstrac- tion of all that is essential in order to prove very general closed graph theorems. Here we concentrate on his theorem in [1] for linear mappings with bornologically closed graph that have an ultrabornological space as domain and a webbed lo- cally convex space as codomain. Henceforth, we shall refer to this theorem as ‘De Wilde’s closed graph theorem’. As far as the category of bounded linear mappings between separated convex bornological spaces is concerned, there exists a corresponding bornological notion of so-called nets that enabled N. Popa in [5] to prove a bornological version of the closed graph theorem which we name ‘Popa’s closed graph theorem’. Although M. De Wilde’s theorem in the locally convex and N. Popa’s theorem in the convex bornological setting are conceptually similar (see also [3]), they do not directly relate to each other. In this manuscript I present a bornological closed graph theorem that generalises the one of N. Popa and even implies the one of M. De Wilde. First of all, I give a suitable definition of bornological webs on separated convex bornological spaces with excellent stability properties and prove the aforementioned general bornological closed graph theorem. It turns out that there is a connection between topological webs in the sense of M.
  • FUNCTIONAL ANALYSIS 1. Metric and Topological Spaces A

    FUNCTIONAL ANALYSIS 1. Metric and Topological Spaces A

    FUNCTIONAL ANALYSIS CHRISTIAN REMLING 1. Metric and topological spaces A metric space is a set on which we can measure distances. More precisely, we proceed as follows: let X 6= ; be a set, and let d : X×X ! [0; 1) be a map. Definition 1.1. (X; d) is called a metric space if d has the following properties, for arbitrary x; y; z 2 X: (1) d(x; y) = 0 () x = y (2) d(x; y) = d(y; x) (3) d(x; y) ≤ d(x; z) + d(z; y) Property 3 is called the triangle inequality. It says that a detour via z will not give a shortcut when going from x to y. The notion of a metric space is very flexible and general, and there are many different examples. We now compile a preliminary list of metric spaces. Example 1.1. If X 6= ; is an arbitrary set, then ( 0 x = y d(x; y) = 1 x 6= y defines a metric on X. Exercise 1.1. Check this. This example does not look particularly interesting, but it does sat- isfy the requirements from Definition 1.1. Example 1.2. X = C with the metric d(x; y) = jx−yj is a metric space. X can also be an arbitrary non-empty subset of C, for example X = R. In fact, this works in complete generality: If (X; d) is a metric space and Y ⊆ X, then Y with the same metric is a metric space also. Example 1.3. Let X = Cn or X = Rn. For each p ≥ 1, n !1=p X p dp(x; y) = jxj − yjj j=1 1 2 CHRISTIAN REMLING defines a metric on X.
  • Uniform Boundedness Principle for Unbounded Operators

    Uniform Boundedness Principle for Unbounded Operators

    UNIFORM BOUNDEDNESS PRINCIPLE FOR UNBOUNDED OPERATORS C. GANESA MOORTHY and CT. RAMASAMY Abstract. A uniform boundedness principle for unbounded operators is derived. A particular case is: Suppose fTigi2I is a family of linear mappings of a Banach space X into a normed space Y such that fTix : i 2 Ig is bounded for each x 2 X; then there exists a dense subset A of the open unit ball in X such that fTix : i 2 I; x 2 Ag is bounded. A closed graph theorem and a bounded inverse theorem are obtained for families of linear mappings as consequences of this principle. Some applications of this principle are also obtained. 1. Introduction There are many forms for uniform boundedness principle. There is no known evidence for this principle for unbounded operators which generalizes classical uniform boundedness principle for bounded operators. The second section presents a uniform boundedness principle for unbounded operators. An application to derive Hellinger-Toeplitz theorem is also obtained in this section. A JJ J I II closed graph theorem and a bounded inverse theorem are obtained for families of linear mappings in the third section as consequences of this principle. Go back Let us assume the following: Every vector space X is over R or C. An α-seminorm (0 < α ≤ 1) is a mapping p: X ! [0; 1) such that p(x + y) ≤ p(x) + p(y), p(ax) ≤ jajαp(x) for all x; y 2 X Full Screen Close Received November 7, 2013. 2010 Mathematics Subject Classification. Primary 46A32, 47L60. Key words and phrases.
  • Boundedness in Linear Topological Spaces

    Boundedness in Linear Topological Spaces

    BOUNDEDNESS IN LINEAR TOPOLOGICAL SPACES BY S. SIMONS Introduction. Throughout this paper we use the symbol X for a (real or complex) linear space, and the symbol F to represent the basic field in question. We write R+ for the set of positive (i.e., ^ 0) real numbers. We use the term linear topological space with its usual meaning (not necessarily Tx), but we exclude the case where the space has the indiscrete topology (see [1, 3.3, pp. 123-127]). A linear topological space is said to be a locally bounded space if there is a bounded neighbourhood of 0—which comes to the same thing as saying that there is a neighbourhood U of 0 such that the sets {(1/n) U} (n = 1,2,—) form a base at 0. In §1 we give a necessary and sufficient condition, in terms of invariant pseudo- metrics, for a linear topological space to be locally bounded. In §2 we discuss the relationship of our results with other results known on the subject. In §3 we introduce two ways of classifying the locally bounded spaces into types in such a way that each type contains exactly one of the F spaces (0 < p ^ 1), and show that these two methods of classification turn out to be identical. Also in §3 we prove a metrization theorem for locally bounded spaces, which is related to the normal metrization theorem for uniform spaces, but which uses a different induction procedure. In §4 we introduce a large class of linear topological spaces which includes the locally convex spaces and the locally bounded spaces, and for which one of the more important results on boundedness in locally convex spaces is valid.
  • On the Second Dual of the Space of Continuous Functions

    On the Second Dual of the Space of Continuous Functions

    ON THE SECOND DUAL OF THE SPACE OF CONTINUOUS FUNCTIONS BY SAMUEL KAPLAN Introduction. By "the space of continuous functions" we mean the Banach space C of real continuous functions on a compact Hausdorff space X. C, its first dual, and its second dual are not only Banach spaces, but also vector lattices, and this aspect of them plays a central role in our treatment. In §§1-3, we collect the properties of vector lattices which we need in the paper. In §4 we add some properties of the first dual of C—the space of Radon measures on X—denoted by L in the present paper. In §5 we imbed C in its second dual, denoted by M, and then identify the space of all bounded real functions on X with a quotient space of M, in fact with a topological direct summand. Thus each bounded function on X represents an entire class of elements of M. The value of an element/ of M on an element p. of L is denoted as usual by/(p.)- If/lies in C then of course f(u) =p(f) (usually written J fdp), and thus the multiplication between Afand P is an extension of the multiplication between L and C. Now in integration theory, for each pEL, there is a stand- ard "extension" of ffdu to certain of the bounded functions on X (we confine ourselves to bounded functions in this paper), which are consequently called u-integrable. The question arises: how is this "extension" related to the multi- plication between M and L.
  • Baire Category Theorem and Uniform Boundedness Principle)

    Baire Category Theorem and Uniform Boundedness Principle)

    Functional Analysis (WS 19/20), Problem Set 3 (Baire Category Theorem and Uniform Boundedness Principle) Baire Category Theorem B1. Let (X; k·kX ) be an innite dimensional Banach space. Prove that X has uncountable Hamel basis. Note: This is Problem A2 from Problem Set 1. B2. Consider subset of bounded sequences 1 A = fx 2 l : only nitely many xk are nonzerog : Can one dene a norm on A so that it becomes a Banach space? Consider the same question with the set of polynomials dened on interval [0; 1]. B3. Prove that the set L2(0; 1) has empty interior as the subset of Banach space L1(0; 1). B4. Let f : [0; 1) ! [0; 1) be a continuous function such that for every x 2 [0; 1), f(kx) ! 0 as k ! 1. Prove that f(x) ! 0 as x ! 1. B5.( Uniform Boundedness Principle) Let (X; k · kX ) be a Banach space and (Y; k · kY ) be a normed space. Let fTαgα2A be a family of bounded linear operators between X and Y . Suppose that for any x 2 X, sup kTαxkY < 1: α2A Prove that . supα2A kTαk < 1 Uniform Boundedness Principle U1. Let F be a normed space C[0; 1] with L2(0; 1) norm. Check that the formula 1 Z n 'n(f) = n f(t) dt 0 denes a bounded linear functional on . Verify that for every , F f 2 F supn2N j'n(f)j < 1 but . Why Uniform Boundedness Principle is not satised in this case? supn2N k'nk = 1 U2.( pointwise convergence of operators) Let (X; k · kX ) be a Banach space and (Y; k · kY ) be a normed space.
  • FUNCTIONAL ANALYSIS 1. Banach and Hilbert Spaces in What

    FUNCTIONAL ANALYSIS 1. Banach and Hilbert Spaces in What

    FUNCTIONAL ANALYSIS PIOTR HAJLASZ 1. Banach and Hilbert spaces In what follows K will denote R of C. Definition. A normed space is a pair (X, k · k), where X is a linear space over K and k · k : X → [0, ∞) is a function, called a norm, such that (1) kx + yk ≤ kxk + kyk for all x, y ∈ X; (2) kαxk = |α|kxk for all x ∈ X and α ∈ K; (3) kxk = 0 if and only if x = 0. Since kx − yk ≤ kx − zk + kz − yk for all x, y, z ∈ X, d(x, y) = kx − yk defines a metric in a normed space. In what follows normed paces will always be regarded as metric spaces with respect to the metric d. A normed space is called a Banach space if it is complete with respect to the metric d. Definition. Let X be a linear space over K (=R or C). The inner product (scalar product) is a function h·, ·i : X × X → K such that (1) hx, xi ≥ 0; (2) hx, xi = 0 if and only if x = 0; (3) hαx, yi = αhx, yi; (4) hx1 + x2, yi = hx1, yi + hx2, yi; (5) hx, yi = hy, xi, for all x, x1, x2, y ∈ X and all α ∈ K. As an obvious corollary we obtain hx, y1 + y2i = hx, y1i + hx, y2i, hx, αyi = αhx, yi , Date: February 12, 2009. 1 2 PIOTR HAJLASZ for all x, y1, y2 ∈ X and α ∈ K. For a space with an inner product we define kxk = phx, xi . Lemma 1.1 (Schwarz inequality).
  • 3 Bilinear Forms & Euclidean/Hermitian Spaces

    3 Bilinear Forms & Euclidean/Hermitian Spaces

    3 Bilinear Forms & Euclidean/Hermitian Spaces Bilinear forms are a natural generalisation of linear forms and appear in many areas of mathematics. Just as linear algebra can be considered as the study of `degree one' mathematics, bilinear forms arise when we are considering `degree two' (or quadratic) mathematics. For example, an inner product is an example of a bilinear form and it is through inner products that we define the notion of length in n p 2 2 analytic geometry - recall that the length of a vector x 2 R is defined to be x1 + ... + xn and that this formula holds as a consequence of Pythagoras' Theorem. In addition, the `Hessian' matrix that is introduced in multivariable calculus can be considered as defining a bilinear form on tangent spaces and allows us to give well-defined notions of length and angle in tangent spaces to geometric objects. Through considering the properties of this bilinear form we are able to deduce geometric information - for example, the local nature of critical points of a geometric surface. In this final chapter we will give an introduction to arbitrary bilinear forms on K-vector spaces and then specialise to the case K 2 fR, Cg. By restricting our attention to thse number fields we can deduce some particularly nice classification theorems. We will also give an introduction to Euclidean spaces: these are R-vector spaces that are equipped with an inner product and for which we can `do Euclidean geometry', that is, all of the geometric Theorems of Euclid will hold true in any arbitrary Euclidean space.
  • POLARS and DUAL CONES 1. Convex Sets

    POLARS and DUAL CONES 1. Convex Sets

    POLARS AND DUAL CONES VERA ROSHCHINA Abstract. The goal of this note is to remind the basic definitions of convex sets and their polars. For more details see the classic references [1, 2] and [3] for polytopes. 1. Convex sets and convex hulls A convex set has no indentations, i.e. it contains all line segments connecting points of this set. Definition 1 (Convex set). A set C ⊆ Rn is called convex if for any x; y 2 C the point λx+(1−λ)y also belongs to C for any λ 2 [0; 1]. The sets shown in Fig. 1 are convex: the first set has a smooth boundary, the second set is a Figure 1. Convex sets convex polygon. Singletons and lines are also convex sets, and linear subspaces are convex. Some nonconvex sets are shown in Fig. 2. Figure 2. Nonconvex sets A line segment connecting two points x; y 2 Rn is denoted by [x; y], so we have [x; y] = f(1 − α)x + αy; α 2 [0; 1]g: Likewise, we can consider open line segments (x; y) = f(1 − α)x + αy; α 2 (0; 1)g: and half open segments such as [x; y) and (x; y]. Clearly any line segment is a convex set. The notion of a line segment connecting two points can be generalised to an arbitrary finite set n n by means of a convex combination. Given a finite set fx1; x2; : : : ; xpg ⊂ R , a point x 2 R is the convex combination of x1; x2; : : : ; xp if it can be represented as p p X X x = αixi; αi ≥ 0 8i = 1; : : : ; p; αi = 1: i=1 i=1 Given an arbitrary set S ⊆ Rn, we can consider all possible finite convex combinations of points from this set.
  • Functional Analysis Lecture Notes Chapter 3. Banach

    Functional Analysis Lecture Notes Chapter 3. Banach

    FUNCTIONAL ANALYSIS LECTURE NOTES CHAPTER 3. BANACH SPACES CHRISTOPHER HEIL 1. Elementary Properties and Examples Notation 1.1. Throughout, F will denote either the real line R or the complex plane C. All vector spaces are assumed to be over the field F. Definition 1.2. Let X be a vector space over the field F. Then a semi-norm on X is a function k · k: X ! R such that (a) kxk ≥ 0 for all x 2 X, (b) kαxk = jαj kxk for all x 2 X and α 2 F, (c) Triangle Inequality: kx + yk ≤ kxk + kyk for all x, y 2 X. A norm on X is a semi-norm which also satisfies: (d) kxk = 0 =) x = 0. A vector space X together with a norm k · k is called a normed linear space, a normed vector space, or simply a normed space. Definition 1.3. Let I be a finite or countable index set (for example, I = f1; : : : ; Ng if finite, or I = N or Z if infinite). Let w : I ! [0; 1). Given a sequence of scalars x = (xi)i2I , set 1=p jx jp w(i)p ; 0 < p < 1; 8 i kxkp;w = > Xi2I <> sup jxij w(i); p = 1; i2I > where these quantities could be infinite.:> Then we set p `w(I) = x = (xi)i2I : kxkp < 1 : n o p p p We call `w(I) a weighted ` space, and often denote it just by `w (especially if I = N). If p p w(i) = 1 for all i, then we simply call this space ` (I) or ` and write k · kp instead of k · kp;w.
  • Representation of Bilinear Forms in Non-Archimedean Hilbert Space by Linear Operators

    Representation of Bilinear Forms in Non-Archimedean Hilbert Space by Linear Operators

    Comment.Math.Univ.Carolin. 47,4 (2006)695–705 695 Representation of bilinear forms in non-Archimedean Hilbert space by linear operators Toka Diagana This paper is dedicated to the memory of Tosio Kato. Abstract. The paper considers representing symmetric, non-degenerate, bilinear forms on some non-Archimedean Hilbert spaces by linear operators. Namely, upon making some assumptions it will be shown that if φ is a symmetric, non-degenerate bilinear form on a non-Archimedean Hilbert space, then φ is representable by a unique self-adjoint (possibly unbounded) operator A. Keywords: non-Archimedean Hilbert space, non-Archimedean bilinear form, unbounded operator, unbounded bilinear form, bounded bilinear form, self-adjoint operator Classification: 47S10, 46S10 1. Introduction Representing bounded or unbounded, symmetric, bilinear forms by linear op- erators is among the most attractive topics in representation theory due to its significance and its possible applications. Applications include those arising in quantum mechanics through the study of the form sum associated with the Hamil- tonians, mathematical physics, symplectic geometry, variational methods through the study of weak solutions to some partial differential equations, and many oth- ers, see, e.g., [3], [7], [10], [11]. This paper considers representing symmetric, non- degenerate, bilinear forms defined over the so-called non-Archimedean Hilbert spaces Eω by linear operators as it had been done for closed, positive, symmetric, bilinear forms in the classical setting, see, e.g., Kato [11, Chapter VI, Theo- rem 2.23, p. 331]. Namely, upon making some assumptions it will be shown that if φ : D(φ) × D(φ) ⊂ Eω × Eω 7→ K (K being the ground field) is a symmetric, non-degenerate, bilinear form, then there exists a unique self-adjoint (possibly unbounded) operator A such that (1.1) φ(u, v)= hAu, vi, ∀ u ∈ D(A), v ∈ D(φ) where D(A) and D(φ) denote the domains of A and φ, respectively.