1 A quick look at topological and functional spaces

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1 1 A quick look t topologicl nd functionl spces The unified chrcter of mthemtics lies in its very nture; indeed, mthemtics is the foundtion of ll exct nturl sciences. Dvid Hilbert ( ) Nowdys, functionl nlysis, tht is minly concerned with the study of complete normed vector spces, occupies centrl plce in modern mthemticl nlysis. Initilly motivted by the understnding nd the study of differentl nd integrl equtions rising in pplied mthemtics, it hs lrgely developed nd evolved round the theory of Bnch nd Hilbert spces nd their rich geometric structure. The importnce nd the verstility of Hilbert spces is exmpled by the spce of Lebesgue squre integrble functions. In this context, most functionl spces hve infinite dimension nd the clssicl theory focusses on liner opertors between these spces. To better understnd the conceptul brekdown in rel nlysis offered by the new functionl spces, we introduce the following exmple, borrowed from [LV02]. Consider for instnce the wve eqution model, simplified model describing the trnsversl oscilltions u = u(x, t) of stretched vibrting string in one dimension of spce 2 u t 2 = c2 2 u x 2 nd supposed to be pegged t its two endpoints, i.e., edowed with the boundry conditions u(0,t)=u(1,t) = 0. The physicl interprettion suggests lso to specify two initil conditions u(x, 0) = u 0 (x), nd u t (x, 0) = v 0(x), which prescribes the initil position of the string nd the initil velocity of the points of the string. The nturl setting for finding solution with finite energy, i.e., such tht 1 0 ( ) 2 u < +, nd t 1 0 ( ) 2 u < +, x Pge: 3 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34

2 4 1 A quick look t topologicl nd functionl spces is to use Fourier series nd then the solution will hve the generl form u(x, t) = ( k cos(kπct)+b k sin(kπct) ) sin(kπx), k=1 nd the initil nd boundry conditions llow to explicit the coefficients k nd b k s k =2 1 0 u 0 (x) sin(kπx)dx, nd b k = 2 kπc 1 0 v 0 (x) sin(kπx)dx. Such solution involves n infinite sum nd is described by denumerble set of coefficients k nd b k. This suggests tht the spce of solution shll be infinite dimensionl. And this sttement showed the limits of the yet known results in rel nlysis. Indeed, the clssicl Bolzno-Weierstrss theorem bout the notion of convergence in finite dimensionl Eucliden spce (i.e., every bounded sequence dmits bounded subsequence in R n ) breks prt. Fortuntely, Hilbert spces were introduced nd provided convenient setting for nlyzing this type of problem. In this chpter we summrize mny of the bstrct concepts, definitions nd theoreticl results on the functionl spces tht re relevnt in functionl nlysis nd importnt for understnding the properties of the solutions of prtil differentil equtions. These bstrct spces, metric spces, normed spces, inner-product spces, re topologicl lgebric spces tht hve ll been introduced in the lst three decdes of the nineteenth century nd the first decdes of the twentieth century 1. They ultimtely led to generliztion of the notions of functions, continuity, differentibility nd integrbility. Until then, functions were ssumed to be continuous, hve derivtives t lmost ll points nd were integrble by existing integrtion methods. In Section 1, we recll the elementry topologicl spces nd the fundmentl properties, seprbility, compctness nd completeness. Section 2, Lesbegue integrtion is introduced s simply s possible, without referring explicitly to the mesure theory, essentilly in view of presenting L p spces. Hilbert spces re the core of Section 3, in which the projection theorem nd Riesz lemm re exposed. In Section 4, distributions re clssiclly discussed in connection with functionl nlysis to show how this generliztion of functions is useful for expressing solutions of prtil differentil equtions. Finlly, Section 5, we introduce Sobolev spces which offer convenient setting for investigting prtil differentil equtions. 1 Students nd persons interested in biogrphicl notes bout the founders of functionl nlysis will red with profit the comprehensive introduction to functionl nlysis by Kren Sxe [Sx01]. Pge: 4 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34

3 1.1 Elementry topologicl spces 1.1 Elementry topologicl spces 5 Before deling with Bnch nd Hilbert spces, we shll first recll some bsic notions nd results from metric nd topologicl spces. The reder must keep in mind tht importnt issues bout prtil differentl equtions concern the notions of convergence (or limits) nd continuity of functions. For instnce, let consider set of points X with notion of distnce between ny two points of X. The convergence of sequence of points (x n ) n 1 X to point x X consists in mesuring the distnce from x n to x nd looking if this distnce tends to 0 s n tends to infinity. There is generl setting for this concept Metric spces The notion of bstrct metric spce is due to M. Fréchet 2 ( ). Definition A metric spce is couple (X, d), where X is set nd d is metric or distnce function on X, i.e., d : X X R + is such tht 1. for ny x, y X, d(x, y) 0 ( non-negtivity) 2. for ny x, y X, d(x, y) =0if nd only if x = y ( identity) 3. for ny x, y X, d(x, y) =d(y, x) ( symmetry) 4. for ny x, y, z X, d(x, z) d(x, y)+d(y, z) ( tringle inequlity). If identity 2 does not hold, then d is then clled semi-metric. Usully, only three conditions re used to define distnce function. Indeed, the first of these conditions is property tht follows from the other three, since: 2d(x, y) =d(x, y)+d(y, x) d(x, x) =0. Furthermore, the inverse tringle inequlity is strightforwrd to obtin d(x, y) d(z, y) d(x, z). In this definition of metric spce, the nture of the elements in the spce is not significnt. For most problems in this textbook, metric spces of functions will be considered, when looking for solutions of prtil differentil equtions. Sequence spces nd function spces In the sequel, we will often del with the following infinite dimensionl metric spces of rel or complex sequences, 1. the metric spce l of ll bounded sequences (x n ) n 1, for the metric d(x, y) = sup x i y i ; i 2 M. Fréchet, Sur quelques points du clcul fonctionnel, Rendic. Circ. Mt. Plermo, 22, 1-74, (1906). Pge: 5 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34

4 6 1 A quick look t topologicl nd functionl spces 2. the metric spce c 0 of ll sequences tht converges to 0 with the metric of l (s we observe tht c 0 l ); 3. the metric spce l p (1 p< ) consisting of ll sequences (x n ) n 1 such tht i=1 x i p <, for the metric d(x, y) = x i y i p i 1 1/p ; In this regrd, the spce l 1 is the spce of ll bsolutely convergent sequences, i.e., (x n ) n 1 is in l 1 if the series i 1 x i converges. Probbly the most importnt mong ll l p -spces is the spce l 2. Let be ny closed, bounded domin in R n. A nturl mesure of the discrepncy between two continuous functions f nd g is given by d(f, g) = sup f(x) g(x), for ll x. (1.1) x The subspce of ll continuous functions on supplied with the metric (1.1) is denoted (C 0 (), ), or simply C 0 (). By extension, the spce of ll continuous functions on closed, bounded domin whose derivtives up to order k re continuous re denoted by C k (). It is metric spce for the distnce function d(f, g) = sup D α f(x) D α g(x), x α k where we introduced the clssicl differentil nottion D α α f f = x α xαn n with α = n α i. (1.2) Let R n be compct domin, Jordn mesurble, we cn consider nother metric on C 0 () i=1 ( 1/p d(f, g) = f(x) g(x) dx) p, (p 1), where d defines metric thnks to the Minkowski inequlity (see Section 1.2) ( 1/p ( ) 1/p ( ) 1/p f(x)+g(x) dx) p f(x) p + g(x) p, p 1. The support of function f : X C n, denoted by supp(f), is the closure of ll points x such tht f(x) 0. Pge: 6 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34

5 1.1.2 The topology of metric spces 1.1 Elementry topologicl spces 7 A specific subset of points in X contining given point x X defines neighborhood of x. Let (X, d) be metric spce nd r stricly positive sclr vlue. The set B r (x) ={y X : d(x, y) <r} is clled the open bll round x with rdius r. A point x U X is clled n interior point of U if U contins some bll round x nd then U is clled neighborhood of x. A point x is clled limit point of U if (B r (x)\{x}) U for every bll round x. A point x is clled n isolted point of U if there exists neighborhood of x not contining ny other point of U. A set U is dense in M if every point of M is limit point of U. The closure of U, denoted by Ū, is the set of points x X such tht ny open bll B r (x) (r>0) contins point of U, i.e., U with its limits points. The interior of U, denoted by U or int(e), is the set of points x X such tht there exists n open bll B r (x) (r>0) which is contined in U, i.e., the set of interior points of U. The set U is bounded if for ech x U, there exists r > 0 such tht U B r (x). The set U is totlly bounded if nd only if, for ny r>0, there exists finite cover (U i ) of U with blls of rdius r (ech set U i in the fmily is of size r or less). In prticulr, we hve the properties 1. U is open if nd only if U = U; 2. U is closed if nd only if Ū = U nd 3. U is dense in X if nd only if Ū = X. A set U contining only interior points is clled open, it is then union of open blls. Its complement is clled closed. A fmily of sets is clled cover of U, if U is contined in the union of these sets. It is n open cover if ech set is open. If C is cover of U, subcover of U is subset of C tht still covers U. A set U is compct if its of ech open cover of U contins finite subcover. A set U is sequentilly compct if every sequence of U contins convergent subsequence. Proposition For ny subset of the Eucliden spce R n, the following four conditions re equivlent 1. Every open cover hs finite subcover. 2. Every sequence in the set hs convergent subsequence, the limit point of which belongs to the set. 3. Every infinite subset of the set hs t lest one ccumultion point in the set. 4. The set is closed nd bounded. In other spces, these conditions my or my not be equivlent, depending on the properties of the spce. The blls generte topology on X, mking it topologicl spce. Pge: 7 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34

6 8 1 A quick look t topologicl nd functionl spces Definition A topologicl spce is couple (X, T ) where X is nonempty set of points nd T is collection of open subsets of X stisfying the xioms 1. The emptyset nd X re in T ; 2. The intersection of ny finite collection of sets in T is lso in T ; 3. The union of ny collection of sets in T is lso in T. The collection T is clled topology on X. If X is ny nonempty set, there re usully different choices for the topology T. Two interesting choices re 1. T = {,X}, clled the trivil topology; 2. T = P (X) (the power set of X) tht consists of the collection of ll subsets of X, clled the discrete topology. Given two topologies T 1 nd T 2 on X, T 1 is clled corser (or weker) thn T 2 if nd only if T 1 T 2, nd T 2 is then finer thn T 1. We consider the following useful topologicl results. Proposition Given metric spce (X, d), the following ssertions hold 1. The sets nd X re both open nd closed. 2. A set O in (X, d) is open if nd only if its complement O c = X\O is closed. 3. Any finite intersection of open sets is open. 4. Any intersection of closed sets is closed. 5. The union of ny finite number of closed sets is closed. 6. If E is compct subset of (X, d), then E is closed Seprbility, compctness nd completeness A metric spce (X, d) is seprble if it contins countble dense subset, i.e. subset with countble number of elements whose closure is the spce itself. Recll tht sequence (x n ) n 1 of elements in X is sid to be Cuchy (or is clled Cuchy sequence) if given ny ε> 0, there exists n integer n 0 such tht for ll m n 0, d(x n,x m ) ε, whenever n. Interestingly, ny convergent sequence is Cuchy. While in R n the converse is true, there exists metric spces in which the converse does not hold. A subset U of (X, d) is clled complete if nd only if every Cuchy sequence in U converges to point of U nd (X, d) is complete if nd only if every Cuchy sequence converges. Hence, we deduce tht sequentilly compct spce is lso complete. Lemm Suppose f C 0 (C) is defined on sequentilly compct metric spce X. Then, the following ssertions hold 1. the modulus of f is bounded nd f ttins its bound. Pge: 8 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34

7 1.1 Elementry topologicl spces 9 2. f is uniformly continuous, i.e., given ε> 0, there exists δ> 0 such tht d(x, y) δ f(x) f(y) ε. We recll n importnt result in Eucliden nd rbitrry metric spces. Theorem (Heine-Borel). A subset U of R n is compct if nd only if it is closed nd bounded. A subset U of metric spce (X, d) is compct if nd only if it is complete nd totlly bounded. nd we cn summrize Corollry (Equivlent forms of compctness). Let (X, d) be metric spce. The following properties re equivlent 1. X is sequentilly compct; 2. X is complete nd totlly bounded; 3. X is compct. Mny of the metrics tht re of interest for this textbook rise from norms Normed spces Definition A normed liner spce over field K (K = R or C) is liner spce V together with mpping R clled norm stisfying 1. for ny v V, v 0, (nonnegtivity) 2. v =0if nd only if v =0, (nondegenercy) 3. for every v V nd λ K, λv = λ v, (multiplictivity) 4. for every v, w V, v + w v + w, (tringle inequlity). Norms give lwys rise to metrics. Indeed, if (V, ) is normed spce, we define metric d on V by posing d(v, w) = v w. Hence, (V, d) is metric spce nd ll notions ssocited to metric spces pply, notbly continuity, compctness nd completness. But notice tht not ll metrics come from norms. Herefter re some bsic exmples of normed spces. 1. V = C with z = z. 2. V = R n with x 2 = x x2 n for x =(x 1,..., x n ). This is the Eucliden norm on R n. Other norms on R n cn be lso defined x 1 = x x n, x = mx 1 i n ( x i ). Pge: 9 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34

8 10 1 A quick look t topologicl nd functionl spces 3. Let be closed, bounded subset of R n. The function f = mx x f(x) defines norm on C 0 (). 4. Let x =(x n ) n 1 l p. The function defined by ( ) 1/p x = x n p (1.3) is norm in l p. Furthermore, we observe tht the Minkowski inequlity is the tringle inequlity for this norm, i.e., given (x n ) n 1, (y n ) n 1 l p, ( ) 1/p x n + y n p ( ) 1/p ( ) 1/p x n p + y n p. Let (V, ) be normed spce. A sequence (x n ) of elements of V converges to x V, if for every ε> 0 there exists n 0 such tht for every n n 0, x n x ε. Consider the spce C 0 () of ll continuous functions defined on closed bounded set R n. A sequence of function (f n ) n 1 is uniformly convergent to f if for every ε> 0 there exists constnt n 0 such tht for ll x nd for ll n > n 0, we hve f n (x) f(x) ε. Notice tht the norm f = mx x f(x) defines the uniform convergence nd is often clled the uniform norm. The normed spce V is imbedded in the normed spce W, nd we write V W if the following conditions re stisfied 1. V is vector subspce of W 2. the identity opertor I defined on V to W by Ix = x for ll x V is continuous. A normed spce is complete if it is complete for the corresponding metric. A complete normed spce is clled Bnch spce. It cn then be checked tht the uniform norm mkes C 0 (C) into Bnch spce. If I R is compct intervl, then C 0 (I) with the mximum norm is Bnch spce. Likewise, l p is Bnch spce for 1 p<, for the norm defined by (1.3). Lemm If S is closed subspce of Bnch spce V nd M is finite dimensionl subspce, then S + M is closed. A liner mp A between two normed spces (V, V ) nd (W, W ) is clled liner opertor. The kernel nd the rnge (or imge) of A re the sets defined s usul by Ker(A) ={f V, Af =0} nd Im(A) ={g W, g = Af}. The opertor A is clled bounded if the opertor norm Pge: 10 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34

9 A = 1.1 Elementry topologicl spces 11 sup Af W f V =1 is finite. The set of ll bounded liner opertors from V to W is denoted by L(V, W ) or L(V ) if V = W. It is normed spce with the opertor norm nd Bnch spce if W is Bnch spce. Lemm An opertor A is bounded if nd only if it is continuous. An opertor in L(V,C) is clled bounded liner functionl nd the spce V = L(V,C) is clled the dul spce of V. The spce V is sometimes denoted by V. Since mny norms rise from inner product, we shll now define this notion Inner product spces Definition Let V be liner vector spce on K. An inner product on V is mpping (, ) :V V K stisfying the conditions 1. (v, v) 0 for ll v V ; (nonnegtivity) 2. (v, v) =0if nd only if v =0; (nondegenercy) 3. (λv, w) =λ(v, w) for ll v, w V nd λ K; (muliplictivity) 4. (v, w) =(w, v) for ll v, w V ; (Hermitin symmetry) 5. (v, w + u) =(v, w)+(v, u) for ll u, v, w V. (distributivity) The inner product is sometimes denoted by, nd my be clled sclr product. A vector spce with n inner product is clled n inner product spce or pre-hilbert spce. Inner products lwys llow to define norms. Indeed, if (V,(, ) is n inner product spce, we define norm on V by posing v = (v, v). Hence, (V, ) is normed spce. Hermitin symmetry nd linerity in the first vrible give s well s (v, λw) =(λw, v) = λ(w, v) = λ(v, w) (v, w + u) =(w + u, v) =(w, v)+(u, v) = (v, w)+(v, u), nd thus n inner product is sesquiliner form. Subsequently, n inner produt on rel vector spce is positive-definite symmetric biliner form. Clerly, the complex liner spce C n with the usul inner product (z, w) = n i=1 z i w i is n inner product spce. Lemm (Cuchy-Schwrz inequlity). If ((V,(, )) is n inner product spce, then for ll v, w V we hve (v, w) v w. Pge: 11 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34

10 12 1 A quick look t topologicl nd functionl spces It is esy to see tht the equlity only occurs if v nd w re coliner. Lemm (Jordn-von Neumnn). A norm is ssocited with inner product if nd only if the prllelogrm rule holds, i.e., 2 v 2 +2 w 2 = v + w 2 + v w 2 Strong nd wek convergence Since every inner product spce is normed spce for the nturlly defined norm x = (x, x), the notion of convergence is well defined. A sequence of vectors (x n ) n 1 in n inner product spce V is sid to converge strongly (or to converge in the norm) to vector v in V if x n x 0 s n. A sequence of vectors (x n ) n 1 in n inner product spce V is sid to converge wekly to vector v in V if (x n,y) (x, y) s n, for ll y in V. Wekly convergent sequences re bounded, i.e., there exists M>0such tht x n M for ll n. The notion of wek convergence defines topology on V tht is clled the wek topology on V. From Cuchy-Schwrzs inequlity, we cn deduce tht the wek topology is weker thn the norm topology. Hence, strongly convergent sequence is lso wekly convergent to the sme limit, while the converse is not true in generl. However, if (x n,x) )x, x) nd x n x, then we hve x n x 0 s n. Complete inner product spces A complete inner product spce is clled Hilbert spce. Next re some interesting inner product nd Hilbert spces 1. if we re considering continuous functions defined over C, C 0 (I), where I =[, b] is compct subset of C, is n inner product spce endowed with the inner product (f, g) = b f(x)g(x)dx nd then the induced norm is then ( 1/2 b f 2 = f(x) dx) 2, nd not the supremum norm f = sup x I f(x). 2. the spce l 2 of complex vlued sequences x =(x n ) n 1 is Hilbert spce endowed with the inner product (x, y) = x n y n. Pge: 12 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34

11 1.2 The Lebesgue integrl 13 A vector v in vector spce V is clled normlized or unit vector if v = 1. Two vectors v, w in vector spce V re orthogonl (v w) if (v, w) = 0 nd prllel if one is multiple of the other. If v nd w re orthogonl, we hve the Pythgoren formul v + w 2 = v 2 + w 2. (1.4) For v nd w 0 in n inner product spce, the projection of v on w is the vector (v, w) ṽ = w, or, if w = 1 ṽ =(v, w)w. w 2 We recll tht set C in vector spce is convex if for ny x, y C, we hve tx + (1 t)t C, with t [0, 1]. Lemm Let C be nonempty convex nd complete liner subspce of inner product spce V. If v V then there exists unique u C minimizing v u nd clled the closest point (or best pproximtion) to v from C. Furthermore, we hve v u C, i.e., (v u, w) =0for ny w C. Given n inner product spce V nd M V, we define the orthogonl spce of M by M = {v V ;(v, m) = 0 for ll m M}. nd this spce M is often referred to s M-perp. Lemm Suppose V is n inner product spce nd M complete liner subspce of V. Then, V = M M, the direct sum being orthogonl. Proposition Suppose V is n inner product spce nd M V. Then M is liner subspce of V, M M nd the intersection M M is either {0} or the emptyset. Before exmining in more detil the properties of Hilbert spces, we like to introduce the Lesbesgue integrl, fundmentl concept for understnding most pplictions Hilbert spce theory. 1.2 The Lebesgue integrl We shll recll tht one purpose of this textbook is to introduce the min numericl methods for solving prtil differentil equtions. Since the solutions to these equtions involve Hilbert spces, it is importnt to show tht ll differentible functions defined on compct intervl I =[, b] belong to Hilbert spce endowed with the inner product (f, g) = b f(x)g(x)dx. Pge: 13 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34

12 14 1 A quick look t topologicl nd functionl spces And precisely, the smllest of such spces is the spce of Lebesgue squre integrble functions on I. Hence, we will briefly review now this concept of Lebesgue integrl Lebesgue integrtion The clssicl Riemnn integrl is suitble for deling with continuous functions defined on bounded subsets of the Eucliden spce R n, or functions with limited number of discontinuities. However, it cnnot hndle discontinuous functions. The generl setting of the mesure theory is suitble for resolving these drwbcks, the integrl is then defined from the notion of size in some set V. Note to the reder. In this textbook, we re minly concerned with integrtion nd not mesure. Moreover, complete description of the mesure theory is beyond the scope of this book, rtly dedicted to undergrdute students. We hve thus decided to introduce the Lebesgue integrl without referring explicitly to concepts like mesure, following more direct pproch pionnered by [DM90]. A reder interested in this topic will consult with profit the books listed in ppendix of this chpter. Hence, grdute students nd mthemticins could quietly skip this section. A rel vlued f defined on R is clled step function if it cn be written s finite liner combintion of chrcteristic functions of semi-open intervls A i =[ i,b i [ R, i.e., f(x) = n α k χ Ak (x), for ll x R, (1.5) k=1 where α k R nd χ A is the chrcteristic function of A, such tht χ Ak (x) =1 if x [ k,b k [ nd χ Ak (x) = 0 otherwise. In this setting, the intervls A k re ssumed to be disjoint nd if we consider besides miniml number of intervls, then the representtion of f is unique 4. It enjoys severl properties, notbly 1. if f, g re step functions, then f + g nd fg re step functions, 2. if f is step function nd α R, then αf is step function, 3. if f is step function, then f is step function. 4. if f, g re step functions, min(f, g) nd mx(f, g) re step functions. 3 Nmed fter the French mthemticin Henri Lebesgue ( ) who introduced the theory of integrtion in his doctorl disserttion, Intégrle, longueur, ire, University of Nncy, (1902). 4 Advnced reders will esily compre this definition with the notion of mesurble simple function hving finite rnge, in the mesure theory. Pge: 14 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34

13 1.2 The Lebesgue integrl 15 The collection of ll step functions is vector spce on R. The derivtive of step function is the Dirc delt function δ(x) = 0, if x 0 nd δ(x) =+ otherwise. We define the integrl of step function s the Riemnn integrl of this function, i.e., n f = α k (b k k )= f(x)dx. k=1 And we observe tht this definition is independent of ny prticulr representtion of f. We hve then Lemm Given f step function whose support is contined in n k=1 A k, where the A k re disjoint semi-open intervls [ k,b k [. If, for ny M > 0 f <M, then n f M (b k k ). k=1 Next, we give two results tht will be helpful for defining the Lebesgue integrl. Theorem Let (f n ) n 1 be non-incresing sequence of nonnegtive step functions such tht lim n f n (x) =0for every x R. Then, f n (x)dx =0. lim n 2. Let (f n ) n 1 nd (g n ) n 1 be two non decresing sequences of step functions. If lim n f n (x) lim n g n (x) for every x R, then g n (x)dx f n (x)dx. lim n lim n And finlly, we introduce the expected definition of the Lebesgue integrl. Definition (Lebesgue integrl). A rel-vlued function f defined on R is clled Lebesgue integrble if there exists sequence of step functions (f n ) n 1 stisfying the following xioms 1. f n (x) dx < ; 2. f(x) = f n (x), for every x R such tht The integrl of f is then defined by f n (x) <. f(x)dx = f n (x)dx. Pge: 15 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34

14 16 1 A quick look t topologicl nd functionl spces In this definition, condition 2 shows tht f is equl to the sum of series t points where the series converges bsolutely. We will show tht the set of ll points where f does not coincide with f n(x) is smll set, clled null set. Hence, the series converges to f(x) for ll x except null set. This introduces the concept of convergence lmost everywhere. The spce of ll Lebesgue integrble functions defined on R is denoted by L 1 (R). And we will observe tht ll Riemnn integrble functions re Lebesgue integrble. The spce L 1 (R) is vector spce nd the function is liner functionl on L 1 (R). Lemm If f L 1 (R) then f L 1 (R) nd we hve f(x)dx f(x) dx. Corollry Given f, g L 1 (R), 1. if f = f n, then f(x) dx f n (x) dx; 2. min(f, g) L 1 (R) nd mx(f, g) L 1 (R). And we hve the following result. Theorem If f L 1 (R), then for t sclr we hve f(x + t) f(x) dx =0. lim t 0 Proof. If f is step function, the results is immedite. Suppose f is n rbitrry Lebesgue integrble function. Then, given ε> 0, if f = n>0 f n there exists n 0 such tht n>n fn 0 < ε/3, nd we hve n 0 n 0 f(x + t) f(x) dx f n (x + t) f n (x) dx + f n (x + t) dx + f n (x) dx n>n 0 n>n 0 n 0 n 0 = f n (x + t) f n (x) dx +2 f n (x) dx n>n 0 n 0 n 0 < f n (x + t) f n (x) dx +2ε/3. As we know tht n 0 f n is step function, we cn then deduce tht Pge: 16 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34

15 1.2 The Lebesgue integrl 17 n 0 n 0 lim f n (x + t) f n (x) dx =0, t 0 nd thus f(x + t) f(x) dx < ε for t sufficiently smll. Interestingly, L 1 (R) cn be considered s Bnch spce, under specific ssumptions, nd norms cn be defined. To this end, we propose now definition for the null function previously evoked. Definition (null function nd null set). 1. An integrble function f is clled null function if f =0. Furthermore, two integrble functions f, g re equivlent if f g is null function. 2. A set X R is clled null set or mesure-zero set, if its chrcteristic function is null function. Under this definition, ny countble set if null set nd countble union of null sets is lso null set. For exmple, the set of rtionl numbers Q is null set with respect to R n, despite being dense in R n. All subsets of R n of dimension smller thn n re null sets in R n. A clssicl exmple of null set which is not countble is the Cntor set [Hl50]. A set is considered null if it is subset of null set Notions of convergence In mesure theory, property hold lmost everywhere, bbrevited.e., if the set of elements for which this property is not stisfied is null set. Hence, if f, g L 1 (R) nd if the set of elements x R for which f(x) g(x) is null set, then f equls g lmost everywhere, i.e., f = g.e. At this point, we cn introduce the equivlence clss of f L 1 (R) s the set of ll functions g L 1 (R) which re equivlent to f, i.e., [f] ={g L 1 (R), f g =0}, nd then consider the spce L 1 (R) of ll equivlence clsses of Lebesgue integrble functions, endowed with the norm [f] = f. Then, the spce (L 1 (R), ) is normed spce. We recll the definition of convergence in normed spce. Definition A sequence of functions (f n ) L 1 (R) converges in norm to f L 1 (R), nd we denote by f n f i.n., if f n f 0. Pge: 17 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34

16 18 1 A quick look t topologicl nd functionl spces According to this definition, if f n f i.n., then f n f i.n. nd f n f. A sequence of function (f n ) L 1 (R) converges to function f L 1 (R) lmost everywhere, denoted by f n f.e., if f n (x) f(x) for every x except null set. We introduce two importnt results regrding L 1 (R) spces. It reltes the notions of convergence in norm nd convergence lmost everywhere. Theorem (Riesz). 1. The spce L 1 (R) is complete; 2. Given sequence (f n ). If f n f in norm, then there exists subsequence (f pn ) of (f n ) such tht f pn f.e. It comes directly tht the limit with respect to the convergence in norm cn be interchnged with the integrtion, thus leding to write lim f n = lim f n. n n And we shll notice here tht this property does not hold when considering the convergence lmost everywhere. The next results illustrte the min difference between Lebesgue integrtion nd other integrtions. Theorem (Lebesgue s monotone convergence). Let (f n ) be nondecresing sequence of nonnegtive integrble functions, i.e., such tht for every k 1, 0 f k (x) f k+1 (x), for lmost every x R. Let f be defined s the pointwise limit of the sequence, f(x) = lim n f n (x). Then f is integrble nd ( ) lim f n = f. n Theorem (Lebesgue s dominted convergence). Let (f n ) be sequence of squre integrble functions converging lmost everywhere to function f. Moroever, suppose there exists squre integrble function g such tht f n g for ll n. Then f is integrble nd f n f i.n., i.e., ( lim n f n ) = Lemm (Ftou s lemm). Let (f n ) be sequence of nonnegtive integrble functions nd let f = lim inf n f n. Then ( ) f lim inf f n. n f. Pge: 18 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34

17 1.2 The Lebesgue integrl Loclly integrble functions Until now, we hve delt with the integrtion over the whole set R, where the integrl f ws ment for + f. However, we need to define the integrtion over bounded intervls. Let I =[, b] be n intervl nd f : I R be function. We denote by b f or f the integrl of f over the intervl [, b]. It I corresponds to the vlue of the integrl of the product fχ [,b], where χ [,b] represents the chrcteristic function of [, b]. According to this definition, b f corresponds to f on [, b] nd zero otherwise. In ddition, we hve the following conventions b f = 1 b f nd f =0. We observe tht if f L 1 (R), then for ny intervl [, b] on R, b f exists. However, the converse my not hold. Definition A loclly integrble function is function f defined on R such tht for ny compct intervl [, b], the integrl b f exists. Under this definition, the spce L 1 (R) is subspce of the spce of loclly integrble functions tht forms vector spce. Lemm Suppose f is loclly integrble function such tht f g for some function g L 1 (R). Then, f L 1 (R) Lebesgue vs. Riemnn integrtion In this section, we ssume the reder to be fmilir with the definition of the Riemnn integrl nd its properties. We briefly summrize this notion in order to introduce nottions. We consider bounded function f defined on the closed, bounded intervl [, b] R. Suppose tht = x 0 <x 1 < <x n = b is prtition of [, b] together with finite sequence of rel t 1,..., t n such tht for ech k 1, x k 1 t k x k. The lower Riemnn sum nd upper Riemnn sum of f with respect to the prtition (x n ) re respectively defined by L n (f) = n m k (x k x k 1 ), nd U n (f) = k=1 n M k (x k x k 1 ), where m k = inf(f(x); x [x k 1,x k ]) nd M k = sup(f(x); x [x k 1,x k ]). A bounded function f defined on [, b] is clled Riemnn integrble if L n (f) =U n (f) nd we denote the Riemnn integrl by b f(x)dx. k=1 Pge: 19 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34

18 20 1 A quick look t topologicl nd functionl spces An interesting property is tht the Riemnn integrl is considered s the limit of the Riemnn sums of function when the size of the prtition (x n ) tends to zero. The definition of the Lebesgue integrl cnnot be seen s generliztion of the Riemnn integrl, but the following result is interesting in this respect. Theorem If f is Riemnn integrble function on [, b] then f is Lebesgue integrble on [, b] nd both integrls coincide. Proof. For every integer n, we consider prtition of [, b] into 2 n subintervls ech of length (b )/2 n. Next, we define g n (x) = 2 n k=1 m k χ [xk 1 x k [(x), nd h n (x) = 2 n k=1 M k χ [xk 1 x k [(x), nd observe tht (g n ) n 1 is n incresing sequence while (h n ) n 1 is decresing sequence. Denoting g = lim n g n (x) nd h = lim n h n (x) leds to conclude tht g nd h re Lebesgue integrble functions such tht, for lmost every x in [, b] g(x) f(x) h(x). It is not diffuclt to deduce tht, for lmost ll x in [, b] lim (h n(x) g n (x)) = h(x) g(x). n Hence, thnks to the monotone convergence theorem, we hve 0 (h g) = lim (h n g n ) = lim h n lim n n n = lim n U n(f) lim n L n(f) =0. And it follows tht g = h.e., thus f is Lebesgue integrble. Moreover b f(x)dx = lim n b g n (x)dx = where the integrl on the left denotes the Riemnn integrl nd the integrl on the right denotes the Lebesgue integrl. A version of the fundmentl teorem for clculus cn be given for the Lebesgue integrl. Theorem (Fundmentl theorem of clculus). If f is Lebesgue integrble on [, b] nd if we define F (x) = then the function F is continuous in [, b] nd differentible.e. Furthermore, F is differentible.e. nd F (x) =f(x) for lmost every x in [, b]. x f b g = b f, g n Pge: 20 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34

19 From this theorem, we cn lso deduce tht the limits 1.2 The Lebesgue integrl 21 F () = lim F (x) nd F () = lim F (x) x + x b exists nd re finite. The following result llows to introduce chnge of vrible in the Lebesgue integrl in similr wy s it is performed with the Riemnn integrl. Lemm (Chnge of vribles). Let g be nondecresing differentible function defined on bounded intervl [, b] such tht g is integrble over [, b]. We pose g() = lim g(x) nd g(b) = lim g(x). x x b + Suppose f is n integrble function over [g(),g(b)]. Then, the product (f g)g is integrble over [, b] nd we hve g(b) g() f(t) dt = b (f g)(t)g (t) dt The Lebesgue mesure on Eucliden spce Now, we re in good condition for defining more generl concepts like mesure sets nd the Lebesgue mesure. The lter is clssicl mnner of ssigning length, re nd volume to subsets of the Eucliden spce. But remember tht not ll sets re mesurble. Definition We introduce the notions of mesurble set nd of mesure s follows 1. set A is clled mesurble if the chrcteristic function of A is loclly integrble function; 2. given mesurble set A. If the chrcteristic function χ A is n integrble function, then the mesure µ(a) of A is defined by µ(a) = χ A, nd we ssign µ(a) = if χ A is not integrble. Null sets re zero-mesure sets, this justify the terminology. More generlly, mesure µ is countbly dditive nonnegtive function µ. Proposition Let (A k ) k 1 be sequence of disjoint mesureble sets. Then A = k 1 A k is mesurble nd we hve ( ) µ(a) =µ A k = µ(a k ). k 1 k 1 Pge: 21 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34

20 22 1 A quick look t topologicl nd functionl spces The integrl over ny mesurble set cn be defined by f = fχ. Definition (Mesurble function). A function f is clled mesurble if there exists sequence of step function (f n ) n 1 such tht f n f.e. Under this definition, every integrble function is mesurble. Furthermore, every loclly integrble function is mesurble. The mesurble functions form vector spce nd the following properties hold 1. if f is mesurble, then f is mesurble; 2. if f, g re mesurble, then f + g, fg re mesurble; 3. if (f n ) n 1 is sequence of mesurble functions, then the functions (inf f n )(x) = inf f n(x) nd (sup f n )(x) = sup f n (x) n 1 (lim inf f n )(x) = sup( inf f n(x)) nd (lim sup f n )(x) = inf (sup f n (x)) j l n j j l n j re mesurble functions. k More Lebesgue spces It is possible to define rbitrry mesure spces nd we will now introduce few other mesure spces. The spce L 2 (R) The spce of ll loclly integrble functions f such tht f 2 L 1 (R) is denoted by L 2 (R). Functions in L 2 (R) re lso clled squre integrble functions. The spce L 2 (R) is vector spce nd the product of two squre integrble functions is lso function of L 2 (R). Furthermore, if we consider the norm defined by f = ( f 2) 1/2, then (L 2 (R), ) is complete normed spce. The spce of squre integrble functions tht vnish outside n intervl [, b] is denoted by L 2 ([, b]). The spces L 1 (R n ) nd L 2 (R n ) We like to consider the Eucliden spce R n s the mesure spce. Given ( k ) 1 k n nd (b k ) 1 k n in R n, with ech k b k, we consider subsets of R n of the form I =[ 1,b 1 ] [ n,b n ]. For subset I, we define m(i) = n (b k k ). k=1 Pge: 22 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34

21 1.2 The Lebesgue integrl 23 And we observe tht m(i) represents the length, re nd volume of I when n =1, 2 or 3, respectively. We hve introduced the notion of step functions s rel-vlued functions tht hve only finite number of elements in their rnge (cf. Definition 1.5). Every such function cn be decomposed s liner combintion of chrcteristic functions. A step function is mesurble if nd only if ech set A i is mesurble set. Such function is then clled simple mesurble function. Every function defined on R n cn be pproximted by simple functions. For step function f, we write nd we define f = f = n α k χ Ik, k=1 n α k m(i k ). k=1 Next we introduce the notion of Lebesgue integrble function, tht expnds nturlly Definition Definition (Lebesgue integrble function on R n ). A rel (or complex) vlued function f defined on R n is clled Lebesgue integrble if there exists sequence of step functions (f n ) n 1 stisfying the followings xioms 1. f n < ; 2. f(x) = f n (x) for every x R n such tht The integrl of f is then defined by f = f n. f n (x) <. The spce of ll Lebesgue integrble functions on R n is denoted by L 1 (R n ). Likewise, we extend Definition s follows. Definition A function f defined on R n is loclly integrble, if for every bounded intervl I the product fχ I is n integrble function. By nlogy with the previous section, set A R n is clled mesurble if the chrcteristic function of A is loclly integrble function. The mesure µ(a) of A is then defined by the vlue of the integrl µ(a) = χ A nd µ(a) = is χ A is loclly integrble but not integrble. Finlly, function f defined on R n is mesurble is there exists sequence of step functions (f n ) n 1 such tht f n f.e. Pge: 23 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34

22 24 1 A quick look t topologicl nd functionl spces The integrl over mesurble set R n is defined by the integrl of the function f on nd 0 everywhere else, f = fχ. The spce of loclly integrble functions f defined on R n such tht f 2 is integrble is denoted by L 2 (R n ). Functions in L 2 (R n ) re clled squre integrble, The spce of squre integrble functions tht vnish outside n subset R n is denoted by L 2 (). The spces L p (R n ) Finlly, for bounded subset R n, we consider the spce L p (), for 1 p<, of ll rel-vlued Lebesgue mesurble functions defined on such tht f p <. And we define, for p =, the spce L () of ll rel-vlued Lebesgue mesurble functions tht re essentilly bounded, i.e., ess sup x f(x) <, where ess sup f(x) = inf{m 0, f(x) M,.e. in }. x We observe tht the spces L p form sequence of embedded spces, i.e., L () L p () L 2 () L 1 (). Actully, the spce L p consists of equivlence clsses of functions, where two functions belong to the sme equivlence clss if they coincide lmost everywhere. For 1 p<, the spce L p is liner spce. For ny 1 p< there exists unique q such tht 1 p + 1 =1, with q =, if p =1. q The number q is clled the Hölder conjugte of p. Lemm (Hólder s inequlity). Suppose 1 < p < nd 1 < q < re Hölder conjugtes. Given f L p () nd g L q () then fg L 1 () nd 1 fg L 1 () = fg f L p () g L q (), p + 1 q =1. Theorem For 1 p<, the spce L p () is normed liner spce, with the norm defined by ( f L p () = ) 1/p f p nd f L () = ess sup f(x). (1.6) x Pge: 24 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34

23 1.3 Hilbert spces 25 In prticulr, the tringle inequlity holds f + g Lp () f L p () + g L p (), tht is clled Minkowski s inequlity in this context. Furthermore, normed liner spce (V, ) is complete if nd only if j 1 f j converges in norm whenever j 1 f j converges. According to this result, the spce L p () is complete nd thus is Bnch spce for the norm L p (), for ny vlue p. The most importnt L p spces for our purposes re the spces where the mesure µ is the Lebesgue mesure on some subset of R n or µ is counting mesure on N. In this cse, we hve lredy introduced these spces, denoted by l p, s the spces of ll bounded sequences (x n ) n 1 stisfying x n p <, with (x n ) n 1 p =( x n p ) 1/p. n 1 We close this section on Lebesgue functions by giving n interesting result. Theorem The step functions re dense in L p (), for ech 1 p<. It remins to be stted tht the spce L 2 () is Hilbert spce nd its norm is induced by n inner product. This cn be seen by writing (f, g) = f g. This spce will be discussed in the next section. 1.3 Hilbert spces The work of Dvid Hilbert ( ) on qudrtic forms in infinitely mny vribles in his study of integrl equtions impelled the theory of Hilbert spces. Their importnce ws first recognized yers lter by John von Neumnn ( ) in his work on unbounded Hermitin opertors nd he is credited for hving developed the modern theory of Hilbert spces. The spce L 2 ([, b]) is Hilbert spce. We hve seen tht L 2 ([, b]) is normed spce, then we still hve to show tht it is complete. Let (f n ) n 1 be Cuchy sequence in L 2 ([, b]), we hve b Cuchy-Schwrz s inequlity yields b ( b f m f n 1 f m f n 2 0 s m, n. b ) 1/2= f m f n ( ) 1/2 b 2 b f m f n 2 Pge: 25 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34

24 26 1 A quick look t topologicl nd functionl spces nd the right-hnd side term tends to 0 s m, n 0. Hence, (f n ) n 1 is Cuchy sequence in L 1 ([, b]) tht converges to function f L 1 ([, b]). This mens tht b f f n 0 s n. Thnks to the Riesz theorem, there exists subsequence (f pn ) convergent to f.e. For ny ε, we obtin, by letting n nd for p m >p n b f pn f 2 ε, by Ftou s lemm. And thus f L 2 ([, b]). Furthermore, we write b f f n 2 b f f pn 2 + b f pn f n 2 < 2ε, for n sufficiently lrge, nd the completeness is chieved Orthonorml bses A bsis of vector spce E is linerly independent subset B of E spnning E, i.e., such tht ny vector x E cn be written s x = n k=1 α kx n, where x k B nd the α k re sclrs. In inner product spces, the resons for which bses re so importnt re twofold. Infinite sums re considered insted of finite liner combintions nd the notion of orthogonlity replces the liner independence property. Let V be Hilbert spce V endowed with n inner product (, ). A sequence (or set of vectors) (v n ) n 1 is clled n orthonorml sequence if (v k,v j ) = δ k,j, for 1 k, j <, where δ jk denotes the Kronecker delt symbol, i.e. δ jk equls one if j = k nd zero otherwise. If the sequence is infinite, then it converges wekly to 0. For exmple, the set of functions f n (x) = exp(inx)/ 2π, for n Z is n orthonorml sequence for the spce L 2 ([ π, π]) endowed with the L 2 inner product (f, g) = π π f(x)g(x) dx. Likewise, the set of Legendre polynomils P n (x) defined by the Rodrigues formul 2n +1 1 d n ( P 0 (x) =1, P n (x) = (x n n! dx n 1) n), n 1, forms n orthonorml system in the spce L 2 ([ 1, 1]) The next result generlizes the Pythgoren formul (1.4). It cn be estblished by induction. Pge: 26 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34

25 1.3 Hilbert spces 27 Lemm (Pythgoren formul). Let (f k ) 1 k n be set of orthogonl vectors in n inner product spce V. Then, n 2 n f k = f k 2. k=1 Suppose (f n ) n 1 is n orthonorml sequence in n inner product spce V. Then, n interesting question would be to find (complex) numbers α n such tht, for ll f V, f = α n f n. Unfortuntely, this cnnot be chieved in generl. Nevertheless, we hve the following results. Lemm Suppose tht (f n ) n 1 is n orthonorml sequence in n inner product spce V nd tht f = α nf n. Then, α n =(f, f n ) for ech n. We cll the term (f, f n)f n the Fourier series of f with respect to the orthonorml sequence (f n ) n 1 nd (f, f n ) re the Fourier coefficients of f with respect to (f n ) n 1. The following result gives detils on the size of these coefficients. Theorem (Bessel s inequlity). Suppose tht (f n ) n 1 is n orthonorml sequence in n inner product spce V. Then, for every f V we hve (f, f n ) 2 f 2. (1.7) In prticulr, this inequlity shows tht the series of nonnegtive numbers (f, f n) 2 converges for every f V. This property mens tht the sequence (f, f n ) n 1 is n element of the Hilbert spce l 2 of squre-summble sequences. Let (f n ) n 1 be n orthonorml sequence in V. If for ny f V, there exists coefficients α n such tht f = k=1 α nf n, then the sequence (f n ) n 1 is clled complete orthonorml sequence in V or n orthonorml bsis for V. According to this definition, n orthonorml sequence (f n ) n 1 in Hilbert spce V is complete if for every f V we hve f = k=1 (f, f n )f n. Actully, this equlity mens tht lim n f (f, f n )f n =0, with respect to the norm in V. Pge: 27 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34

26 28 1 A quick look t topologicl nd functionl spces Theorem (Prsevl s theorem). Suppose tht (f n ) n 1 is n orthonorml sequence in n inner product spce V. Then, (f n ) n 1 is complete orthonorml sequence if nd only if for every f V we hve (f, f n ) 2 = f 2. (1.8) The completeness of V is in generl sufficient to ensure the convergence of the series (f, f n) 2 s stted next. Theorem Let (f n ) n 1 be complete orthonorml sequence in Hilbert spce V nd let (α n ) n 1 be sequence of rel or complex numbers. Then, the series α nf n converges if nd only if α n 2 <. Moreover, in tht cse we hve 2 α n f n = α n 2. And we hve n importnt chrcteriztion of complete orthonorml sequences. Lemm An orthonorml sequence (f n ) n 1 in Hilbert spce V is complete if nd only if (f, f n )=0, for ll n 1 f =0. A Hilbert spce is clled seprble if it contins complete orthonorml sequence. Finite dimensionl Hilbert spces re seprble. If V is seprble, the construction of n orthonorml bsis is esy. Indeed, there exists countble totl set (f n ) n 1. The Grm-Schmidt orthogonliztion procedure (cf. Section 4.2.5) llows to construct n orthonorml set (u n ) n 1 such tht spn(u n ) = spn(f n ) for ny n. Theorem Every seprble inner product spce hs countble orthonorml bsis. Corollry If V is seprble, then every orthonorml bsis is countble. A bijective liner opertor A L(V, W ), where V nd W re Hilbert spces, is clled unitry if A preserves the inner products (or the norms) (Ag, Af) =(g, f), g, f V. Hence, V nd W re clled unitrily equivlent. Let V be n infinite dimensionl Hilbert spce, nd let (f n ) n 1 be ny orthogonl bsis. The mp A : V l 2 (N), f ((f k,f)) k 1 is unitry. Lemm Any seprble infinite dimensionl Hilbert spce is unitrily equivlent to l 2 (N). Pge: 28 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34

27 1.3.2 The projection theorem nd Riesz lemm 1.3 Hilbert spces 29 In this section, we consider closed vector subpces of Hilbert spce V, tht re Hilbert spces since closed subspce of complete normed spce is complete. Let S be nonempty subset of Hilbert spce V. We recll tht u H is orthogonl to S if (u, v) = 0 for every v S. Then, the orthogonl complement of S is the set of ll elements in V orthogonl to S, denoted by S. By continuity of the inner product, it follows tht S is closed liner subspce nd by linerity tht spn(s) = S. Obviously, {0} = V nd V = {0}. Moreover, is S is closed subspce of V, we hve S = S. Lemm Let S be subspce of Hilbert spce V. Then S is dense if nd only if S = {0}. A fundmentl property of Hilbert spces is tht the distnce of point to closed convex set if lwys ttined. This result is especilly importnt in the pproximtion theory. Theorem (Closest point). Let C be closed convex subset of Hilbert spce V. For every element g V, there exists unique closest element (or best pproximtion) f C to g minimizing g f, i.e. such tht g f = inf g h. h C Proof. We prove first the existence of such point. Let (f n ) n 1 be sequence in C such tht lim g f n = inf g h. n h C Posing d = inf h C g f n, we know tht 1 2 (f m + f n ) C nd thus we hve g 1 2 (f m + f n ) d, for ll m, n 1. From the prllelogrm formul, we show tht f m f n 2 =2 g f m 2 +2 g f n 2 4 g 1 2 (f m f n ) 2, nd since 2 g f m 2 +2 g f n 2 tends to 4d 2 when n, m, we conclude tht f m f n 2 tends to 0 when m, n nd thus, (f n ) n 1 is Cuchy sequence. The limit f = lim n f n exists in C since V is complete nd C is closed. We hve then g f = g lim n f n = lim n g f n = d. The uniqueness of f cn be esily obtined by contrdiction. Pge: 29 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34

28 30 1 A quick look t topologicl nd functionl spces This theorem gives n existence nd uniqueness result which is often used in optimiztion problems. However, it is of limited prcticl usefulness for finding this optiml point. The following theorem provides useful chrcteriztion. Corollry Let C be closed convex subset of rel Hilbert spce V. For f C nd g V, the following ssertions re equivlent 1. g f = inf h C g h 2. (g f, h f) 0, for ll h C. Theorem (Projection theorem). Let S be closed subspce of Hilbert spce V. Then every g V cn be uniquely decomposed s g = f + h where f S nd h S, nd thus we write symboliclly H = S S. The spce V is the direct sum of S nd its orthogonl complement S. In other words, to every g V, we cn ssign unique element f which is the element in S closest to f. This property llows us to consider the opertor P s g = f clled the orthogonl projection corresponding to S. Note tht we hve P 2 S = P S, nd (P S f, g) = (f, P S g), for every f, g V. Clerly, we hve lso P S g = g P S g = h, with h S Bounded liner opertors on Hilbert spces Liner opertors on normed nd Hilbert spces ply n importnt role in pplied mthemtics. We turn now to liner functionls A : V C nd we will consider lso biliner functionls nd qudrtic forms. We recll tht liner opertor A : X Y, where X, Y re normed liner spces, is clled bounded if the opertor norm is finite, i.e., if there exists M>0 such tht Af Y M f X, for ll f X. A consequence of the linerity of the opertor is tht continuity cn be checked t single point only. Lemm Consider liner opertor A : X Y, where X, Y re normed liner spces. The opertor A is continuous t every point if it is continuous t single point. And ccording to Lemm 1.1.3, liner opertor A : X Y between two normed liner spces is continuous if nd only if it is bounded on X. One of the most importnt opertor for our purposes is obviously the differentil opertor, defined on the spce of differentible functions on n intervl [, b] R, by Pge: 30 job: book mcro: svmono.cls dte/time: 11-Jun-2009/15:34