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<p> </p><p>Preliminaries<br>Throughout the work, K will hold either for the eld of real numbers R or for<br>the eld of complex numbers C.<br>This part is devoted to a review of some basic notions from Functional<br>Analysis and to the introduction of bilinear forms.. The ideal spaces on which<br>we shall work are Hilbert spaces (e.g. euclidean spaces) that are particular<br>(and even pratical) Banach spaces.<br>1.1 Banach spaces<br>Denition 1.1.1<br>Let X be a linear space over K. A norm on X is any nonnegative real-valued<br>function jj jj on X satisfying the following conditions :<br>i) 8 x 2 X, jjxjj = 0 () x = 0: (nondegeneracy)<br>ii) jjxjj = jj jjxjj ; 8 x 2 X and 8 2 K: (homogeneity)<br>iii) jjx + yjj jjxjj + jjyjj, 8 x; y 2 X : (subadditivity)<br>A linear space X endowed with a norm jj jj is called a normed linear space<br>and is denoted by (X; jj jj).<br>A normed linear space (X; jj jj) of which norm is not ambiguous will be<br>simply denoted by X.<br>Denition 1.1.2<br>Let X be a K-linear space. Two norms jj jj1 and jj jj2 on X are said to be<br>equivalent if there exist positive constants and such that<br>jjxjj1 jjxjj2 jjxjj1 8 x 2 X :<br>Denition 1.1.3<br>5<br>Let (X; jj jj) be a normed linear space. A sequence (xn)n of elements of X<br>converges in (X; jj jj), if there exists an element a 2 X such that<br>lim<br>n!+1<br>jjxn ô€€€ ajj = 0 in R:<br>In this case a is unique (due to the triangular inequality property of the<br>norm) and we also say that (xn)n converges to a with respect to the norm jj jj<br>and then write<br>lim<br>n!+1<br>xn = a :<br>Denition 1.1.4<br>Let (X; jj jj) be a normed linear space.<br>1. A subset F of X is said to be closed if every sequence (an)n of elements<br>of A which converges to some element x 2 X, has its limit x in A. That<br>is:<br>(an)n A converges in X =) lim<br>n!+1<br>an 2 A:<br>2. A subset U of X is said to be open if its complement; Uc = X n U, is<br>closed.<br>3. The collection = of all open sets of the normed linear space (X; jj jj)<br>denes a topology on X. In this topological space (X; =), = is called the<br>norm topology.<br>4. The closure of a subset A of X is the smallest closed set (of X) that<br>contains A.<br>The closure of A is in fact the intersection of all closed sets of X containing<br>A.<br>It is denoted by A or cl(A).<br>A subset D X of which closure is equal to X, is said to be dense in<br>X.<br>5. The interior of a subset A of X is the largest open set (of X) which is<br>contained in A.<br>The interior of A is in fact the union of all open sets contained in A.<br>It is denoted by A or int(A).<br>6. A subset A X is said to be bounded if there exists a positive constant<br>M such that<br>x 2 A =) jjxjj &lt; M :<br>This means that A is contained in some open ball B(0; M).<br>Now we dene the most important topological concept for computational<br>purposes, namely the concept of separability.<br>6<br>Denition 1.1.5<br>A normed linear space (X; jj jj) is said to be separable if it contains a dense<br>subset D which is at most countable.<br>Proposition 1.1.6<br>Let X be a K-linear space and assume that jj jj1 and jj jj2 on X are two<br>equivalent norms on X. Then a sequence of elements of X converges with<br>respect to jj jj1 if and only if it does with respect to jj jj2 :<br>Therefore two equivalent norms dene the same topology on X:<br>Vice-versa if two norms dene the same topology, then they are equivalent.<br>However two metrics may dene the same topology without being strongly equiv-<br>alent!<br>Denition 1.1.7<br>Let (X; jj jj) be a normed linear space. A sequence (xn)n of elements of X is<br>said to be a Cauchy sequence if<br>lim<br>m;n!+1<br>jjxn ô€€€ xmjj = 0 ; that is,<br>8″ &gt; 0; 9N 2 N : jjxn ô€€€ xmjj &lt; ” for all n N and all m N :<br>Proposition 1.1.8.<br>In a normed linear space,<br>1) every convergent sequence is a Cauchy sequence,<br>2) and every Cauchy sequence is bounded.<br>Denition 1.1.9 (Banach space)<br>A normed linear space (X; jj jj) in which every Cauchy sequence is convergent<br>is called a Banach space.<br>Denition 1.1.10<br>Given a normed linear space (X; jj jjX), any Banach space (E; jj jjE) such<br>that there exists an isometry j : X ! E with dense range in E; i.e.,<br>(i) jjj(x)jjE = jjxjjX for all x 2 X, and<br>(ii) j(X) = E,<br>7<br>is called a completion of X.<br>This means that E is a completion of X if E is a Banach space which contains<br>a dense subset isometric to X.<br>For instance :<br>i) The completion of C([0; 1]) equipped with the norm jj jj2 dened by<br>jjfjj2 =<br>hR 1<br>0 jf(t)j2dt<br>i1<br>2 , is (isometric to) L2(]0; 1[).<br>ii) The completion of C1([0; 1]) equipped with the norm jj jjW1;2 dened by<br>jjfjjW1;2 =<br>hR 1<br>0 jf(t)j2dt +<br>R 1<br>0 jf0(t)j2dt<br>i1<br>2 , is (isometric to) H1(]0; 1[).<br>Theorem 1.1.11 (Hausdor)<br>Every normed linear space has a completion.<br>Denition 1.1.12<br>Let (X; jj jjX) and (Y; jj jjY ) be two arbitrary normed linear spaces. A map<br>f : X ! Y is said to be continuous at a point a 2 X if for every sequence<br>(xn)n of X converging to a with respect to jj jjX, the sequence<br>ô€€€<br>f(xn)</p><p>n converges<br>to f(a) in Y with respect to jj jjY .<br>f is said to be continuous (on X) if it is continuous at every point of X.<br>Equivalently, f is continuous if and only if the pre-image of every open set in<br>Y is an open set in X.<br>Recall that given two K-linear spaces X and Y ,<br>a map or operator<br>T : X ô€€€! Y<br>is said to be linear if for all x1; x2 2 X and for all 1; 2 2 K we<br>have<br>T(1×1 + 2×2) = 1T(x1) + 2T(x2) :<br>Equivalently, T : X ô€€€! Y is linear if for all x1; x2 2 X and for all<br>2 K, we have<br>T(x1 + x2) = T(x1) + T(x2) :<br>a map or operator<br>T : X ô€€€! Y<br>8<br>is said to be antilinear if for all x1; x2 2 X and for all 1; 2 2 K<br>we have<br>T(1×1 + 2×2) = 1T(x1) + 2T(x2) :<br>Equivalently, T : X ô€€€! Y is linear if for all x1; x2 2 X and for all<br>2 K, we have<br>T(x1 + x2) = T(x1) + T(x2) :<br>Theorem 1.1.13<br>Let (X; jj jjX) and (Y; jj jjY ) be normed linear spaces. Then a linear map<br>T : X ! Y is continuous if and only if T is a bounded linear map in the sense<br>that there exists a constant real number 0 such that<br>jjT(x)jjY jjxjjX 8x 2 X:<br>Notations 1.1.14<br>Let X and Y be two given arbitrary normed linear spaces.<br>The set of all bounded linear maps (i.e. continuous linear maps) from X into<br>Y is a linear space that will be denoted by B(X; Y ). When X = Y , we<br>simply write B(X) instead of B(X;X).<br>Given a bounded linear map T : X ! Y , we shall set<br>jjTjjB(X;Y ) = inf<br>n<br>k : jjT(x)jjY kjjxjjX 8 x 2 X<br>o<br>that will be simply written as jjTjj when there is no ambiguity.<br>We denote by X := B(X;K) the topological dual of X; that is, the set of all<br>continuous linear functionals of X.<br>Elements of X are also called continuous linear forms or bounded linear<br>forms.<br>Proposition 1.1.15<br>Let (X; jj jjX) be a nontrivial normed linear space and (Y; jj jjY ) be an<br>arbitrary normed linear space. Then for every T 2 B(X; Y ), we have<br>jjT(x)jjY jjTjj jjxjjX 8 x 2 X ;<br>and<br>jjTjj = sup<br>jjxjjX1<br>jjT(x)jjY = sup<br>jjxjjX=1<br>jjT(x)jjY = sup<br>jjxjjX6=0<br>jjT(x)jjY<br>jjxjjX<br>:<br>9<br>Theorem 1.1.16<br>Let (X; jj jjX) and (Y; jj jjY ) be normed linear spaces. Then<br>1.<br>ô€€€<br>B(X; Y ) ; jj jjB(X;Y )</p><p>is a normed linear space.<br>2. If moreover (Y; jj jjY ) is a Banach space, then<br>ô€€€<br>B(X; Y ); jj jjB(X;Y )</p><p>is<br>a Banach space.<br>Corollary 1.1.17<br>The dual X of any normed linear space X is (always) a Banach space.<br>Remark 1.1.18<br>Given a normed linear space (X; jj jjX), the dual X being a normed linear<br>space (in fact a Banach space) has also a dual X called the bidual of X.<br>Moreover there exists a canonical injection J : X ,! X dened by<br>J : X ô€€€! X<br>x 7ô€€€! J(x) ;<br>where J(x) is the continuous form on X dened by<br>hJ(x); fi := hf; xi := f(x) ; 8 f 2 X :<br>Denition 1.1.19 (Re exive space)<br>A normed linear space (X; jj jjX) is re exive if it is a Banach space such that<br>the canonical injection J : X ,! X is surjective.<br>Theorem 1.1.20 (Hahn-Banach)<br>Let (X; jj : jj) be a normed linear space and E X be a linear subspace of X:<br>If f : E ô€€€! R is a continuous linear functional, then there exists F 2 X that<br>extends f such that<br>jjFjjX = jjfjjE :<br>Corollary 1.1.21<br>For every x0 2 X, there exists f0 2 X such that<br>jjf0jj = jjx0jj and hf0; x0i = jjx0jj2:<br>10<br>Theorem 1.1.22 (Uniform Boundedness Principle)<br>Let (X; jj jjX) and (Y; jj jjY ) be two Banach spaces and let fTigi2I be a<br>family (not necessarily countable) of continuous linear operators from X into<br>Y . Assume that<br>sup<br>i2I<br>jjTi(x)jj &lt; 1 8x 2 X:<br>Then<br>sup<br>i2I<br>jjTijjB(X;Y ) &lt; 1:<br>In other words there exists a constant c &gt; 0 such that<br>jjTi(x)jj cjjxjj 8x 2 X; 8i 2 I:<br>Theorem 1.1.23 (Open Mapping Theorem)<br>Let (X; jj jjX) and (Y; jj jjY ) be two Banach spaces and let T be a continuous<br>linear operator from X into Y that is surjective. Then there exists a constant<br>c &gt; 0 such that<br>BY (0; c) T(BX(0; 1)):<br>Corollary 1.1.24 (Banach Isomorphism Theorem)<br>Let (X; jj jjX) and (Y; jj jjY ) be two Banach spaces and let T be a continuous<br>linear operator from X into Y that is bijective. Then the inverse map Tô€€€1 is<br>also continuous ( from Y into X).<br>Theorem 1.1.25 (Closed Graph Theorem)<br>Let (X; jj jjX) and (Y; jj jjY ) be two Banach spaces and let T be a linear<br>operator from X into Y: Assume that the graph of T, G(T), is closed in EF:<br>Then T is continuous.<br>Remark 1.1.26<br>The converse is obviously true, since the graph of any continuous map (linear<br>or not) is closed.<br>Denition 1.1.27 (Compactness)<br>A subset A of a Banach space X is said to be compact if every cover of A by<br>open sets of X, has a nite subcover.<br>Proposition 1.1.28<br>Every compact set of a Banach space is closed and bounded.<br>The converse is not true in general.<br>11<br>Theorem 1.1.29 (Bolzano-Weierstrass)<br>A subset A of a Banach space X is said to be compact if and only if any<br>sequence of A has a subsequence that converges to some point of A.<br>Theorem 1.1.30 (Heine-Borel)<br>Given a natural number n, a subset of the euclidean space Rn is compact if<br>and only if it is closed and bounded.<br>Heine-Borel Theorem fails in innite dimensional Banach spaces. In fact we<br>have the following characterization of nite dimensional normed spaces over<br>R.<br>Theorem 1.1.31 (Riesz)[19]<br>Let E be a Banach space over the eld K. Then the closed unit ball of E is<br>compact if and only if the dimension of E is nite.<br>Thus Riesz Theorem characterizes the compactness of the closed unit ball<br>of a Banach space E by the niteness of the dimension of E.<br>Therefore, we need other types of topologies on innite dimensional spaces<br>Denition 1.1.32<br>Let E be a real Banach space. To each f in E, we assign the map<br>f : E ô€€€! R<br>x 7ô€€€! f (x) = hf; xi:<br>We denote the family of all such maps from E into R by ffgf2E .<br>The weak topology on E (denoted by !) is the smallest topology on E which<br>makes the maps f continuous.<br>Proposition 1.1.33<br>Let (E; !) be a real Banach space endowed with the weak topology ! . Then<br>! is Hausdor; that is, for any two dierent points x1 and x2 taken in (E; !);<br>there exists two respective disjoint weakly open neighbourhoods U1 and U2;<br>that is U1, U2 belongs to ! and U1 U2 = ;:<br>Consequently if a sequence fxngn2N in (E; !) converges weakly to some x<br>(i.e xn * x) in (E; !), then x is unique.<br>12<br>Proposition 1.1.34<br>A sequence fxngn2N in a real Banach space E converges weakly to some x in<br>E if and only if f(xn) ô€€€! f(x) for each f 2 E:<br>Proposition 1.1.35<br>Given a nite dimensional normed space E, the norm-topology (strong topology)<br>and the weak topology coincide on E.<br>Theorem 1.1.36 (Eberlein-Smulyan)<br>A real Banach space E is re exive if and only if every (norm) bounded sequence<br>in E has a subsequence which converges weakly to an element of E.<br>On the dual of a normed space, we can dene a weaker topology as follows.<br>Denition 1.1.37<br>Let E be a real Banach space. Dene for every x 2 E, the map ‘x dened on<br>E by<br>‘x(f) = f(x) ; 8 f 2 E :<br>Then the weak topology on E is dened as the smallest topology on E for<br>which the maps ‘x; x 2 E, are continuous.<br>The weak topology of E (denoted by !) is the<br>Proposition 1.1.38<br>Let E be a Banach space. A sequence of bounded linear forms ffngn2N of<br>elements of E converges weakly to some f 2 E if and only if fn(x) ô€€€! f(x)<br>for each x 2 E:<br>Proposition 1.1.39<br>Given a nite dimensional normed space E, the norm-topology (strong topology),<br>the weak topology and the weak topology coincide on East.<br>13<br>Theorem 1.1.40 (Banach-Alaoglu)[19]<br>For every Banach space E, the closed unit ball is weakly compact.<br>Now we consider the spectral properties of bounded linear operators.<br>Denitions 1.1.41 (Spectra and resolvents)<br>Let E be a Banach space over K and T be a bounded linear operator of E;<br>i.e., T 2 B(E). The spectrum (T) of T is dened by<br>(T) = f 2 K : I ô€€€ T is not invertible in B(E)g :<br>The resolvent set (T) of T is the complementary of (T) in K; that is,<br>(T) = K n (T):<br>The elements of (T) are called the regular values of T.<br>If 2 (T), then the operator R(T) = (I ô€€€ T)ô€€€1 is called the resolvent<br>operator of T at .<br>The spectrum is decomposed into the disjoint union of the following three<br>sets:<br>a) The point spectrum of T :<br>p(T) =<br>n<br>2 K : Ker(I ô€€€ T) 6= f0g<br>o<br>:<br>b) The continuous spectrum of T :<br>c(T) =<br>n<br>2 K : Ker(Iô€€€T) = f0g; R(I ô€€€ T) = E but R(Iô€€€T) 6= E<br>o<br>:<br>c) The residual spectrum of T :<br>r(T) =<br>n<br>2 K : Ker(I ô€€€ T) = f0g; R(I ô€€€ T) 6= E<br>o<br>:<br>Remark/Denition 1.1.42 [24],[19]<br>1. Given T 2 B(E), an element of p(T) is called an eigenvalue of T and a<br>non-zero vector v such that Tv = v is called an eigenvector of T associated<br>to the eigenvalue . Eigenvalues and eigenvectors are sometimes<br>called characteristic values and characteristic vectors respectively.<br>More generally if 2 p(T), n 2 N and v is a nonzero element of E such<br>that (I ô€€€ T)nv = 0, then v is called a principal vector associated with<br>the eigenvalue .<br>14<br>2. Given an eigenvalue of an operator T, the geometric multiplicity of is<br>by denition the dimension of Ker(Iô€€€T). And the algebraic multiplicity<br>of is dened as follows:<br>when E is nite-dimensional and equipped with a basis, it is the multiplicity<br>of as a root of the characteristic polynomial of the matrix<br>associated to T,<br>more generally for an arbitrary nontrivivial Banach space E, it is the<br>dimension of the vector subspace<br>1[<br>k=1<br>Ker(I ô€€€ T)k :<br>Thus the algebraic multiplicity of an eigenvalue is always greater<br>than or equal to its geometric multiplicity.<br>The notion of spectrum can be dened for unbounded operators dened<br>in a normed space.<br>Example 1.1.43 [24],[30]<br>1. Let n be a natural number and A : Cn ! Cn be a linear operator with<br>corresponding matrix A. then it is well-known that T has n eigenvalues<br>1; : : : ; n (counted with their multiplicities) and that the eigenvalues<br>are the roots of the characteristic polynomial P() = det(Aô€€€ In) for<br>A. Since A ô€€€ In) is invertible when is not an eigenvalue, it follows<br>that the spectrum (A) of A is a pure point spectrum, namely<br>(A) =</p><p>k : 1 k n</p><p>= p(A) ;<br>and that the resolvent set of A is the complex plane except nitely many<br>points, namely<br>(A) = C n</p><p>k : 1 k n</p><p>:<br>2. Let E = C([0; 1]) be equipped with the supremum norm and consider the<br>operator T : E ô€€€! E; f 7ô€€€! Tf dened by<br>[Tf](x) =<br>Z x<br>0<br>f(s) ds ; 8x 2 [0; 1] :<br>Then it is not hard to check that T has no eigenvalue, T is injective<br>but not surjective and its range is not dense in E, and moreover TI is<br>invertible in B(E). Therefore<br>(T) = f0g = r(T) :<br>15<br>Theorem 1.1.44 [20]<br>Let E be a Banach space over C and T 2 B(E). Then the following holds for<br>the spectrum of T.<br>i) (T) is a closed subset of C<br>ii) (T) B(0; jjTjjB(X))<br>iii) (T) is a compact subset of C<br>Denition 1.1.45 (Spectral radius) [20]<br>Let E be a Banach space over C and T 2 B(E). Then the spectral radius of T<br>is dened by<br>r(T) := sup<br>n<br>jj : 2 (T)<br>o<br>= max<br>n<br>jj : 2 (T)<br>o<br>:<br>Futhermore, we have the Gelfang formula<br>r(T) = lim<br>n!1<br>kTnk<br>1<br>n = inf<br>n2N<br>kTnk<br>1<br>n :<br>Denition 1.1.46 (Compact linear maps)<br>Let E and F be two Banach Spaces over K. A linear map, T : E ! F is said<br>to be compact if the image of the closed unit ball BE(0; 1) by T is a relatively<br>compact subset of F.<br>In other words, T is compact if T<br>ô€€€<br>BE(0; 1)</p><p>is compact.<br>This denition is equivalent to each of the following properties.<br>i) For each bounded subset B E, the image set T(B) is relatively compact<br>in F.<br>ii) For every bounded sequence fxngn2N E, the sequence fTxngn2N has a<br>convergent subsequence in F:<br>Proposition 1.1.47 (Properties of compact linear maps) [20]<br>1. Every compact linear map is bounded.<br>2. For all Banach spaces E and F over the eld K, the set K(E; F) of<br>compact linear maps from E into F is a linear subspace of the space<br>B(E; F) of bounded linear maps from E into F.<br>3. Let E, F and G be Banach spaces, and let T : E ô€€€! F, S1 : F ô€€€! G<br>and S2 : G ô€€€! E be bounded linear operators. Then<br>16<br>i) If the range of T; R(T), is nite dimensional, then T is compact.<br>Therefore, every linear map dened on a nite dimensional normed<br>space is not only bounded, but also compact.<br>ii) If T is compact, then S1 T and T S2 are compact.<br>Therefore the set K(E) of compact linear operators (endomorphisms)<br>of E is a two-sided ideal of the algebra B(E) of bounded linear operators<br>(endomorphisms) of E.<br>iii) The limit of a convergent sequence of compact linear operators with<br>respect to the operator norm, is compact. Thus K(E) in closed in<br>B(E).<br>Next, we state the Riesz-Schauder spectral theory of compact linear operators.<br>Theorem 1.1.48 (Riesz-Schauder spectral theory)[19],[22]<br>Let E be a complex Banach space and T be a compact linear operator of E.<br>Then:<br>1. The spectrum of T consists of an at most countable set of points of the<br>complex plane which has no point of accumulation except possibly = 0.<br>2. Every nonzero number of the spectrum of T is an eigenvalue of T of nite<br>multiplicity.<br>3. The dual operator of T denoted by T 2 B(E) and dened by<br>T(f) = f T for every f 2 E ;<br>is also compact and a nonzero number is an eigenvalue of T if and only<br>if it is an eigenvalue of T.<br>Remark 1.1.49 [22]<br>The notion of dual operator is an extension of the notion of transposed matrix.<br>In fact, if E is nite dimensional and equipped with a given basis, then the<br>matrix of the dual of a linear operator of E is the transposed matrix of the<br>matrix of T. And More generally, if E is a normed space and T 2 B(E), then<br>using the duality pairing of E E, we have<br>hf; Txi = hT f; xi 8 x 2 E :<br>The above Theorem 1.1.48 can be rephrased as follows.<br>17<br>Theorem 1.1.50<br>Let E be a complex Banach space and T be a compact linear operator of E.<br>1. If 2 (T) and 6= 0, then is an eigenvalue of T of nite multiplicity.<br>2. (T) is either nite or countably innite.<br>3. If (T) is innite, then<br>(T) = f0g [</p><p>n : n = 1; 2; : : :</p><p>;<br>where fngn2N is a sequence of complex numbers converging to 0.<br>As a corollary we have<br>Theorem 1.1.51 [20]<br>Let E be an innite dimensional complex Banach space and T 2 B(E) be<br>compact. Then The following holds.<br>1. 0 2 (T)<br>2. (T) n f0g consists of eigenvalues of nite multiplicity.<br>3. (T) n f0g is either empty, nite or a sequence of complex numbers<br>converging to 0. That is (T) n f0g is a discrete set with no limit point<br>other than 0.<br>1.2 Hilbert spaces<br>Denition 1.2.1 (Inner product)<br>An inner product on a K-linear space E is any functional h ; i dened on EE<br>which is a positive hermitian and nondegenerate form; that is,<br>h ; i : E E ô€€€! K<br>(x; y) 7! hx; yi<br>and satises the following conditions :<br>1. hx; xi 0 8 x 2 E, and hx; xi = 0 if and only if x = 0.<br>2. hy; xi = hx; yi 8 x; y 2 E.<br>3. h1x1 + 2×2; yi = 1hx1; yi + 2hx2; yi 8 x1; x2; y 2 E<br>and 8 x1; 2 2 K.<br>18<br>Remark 1.2.2<br>1. A form : E E ! K which is linear with respect to its rst argument<br>and antilinear with respect to its second argument is said to be<br>sesquilinear.<br>If : E E ! K is sesquilinear and satisfy<br>(x; y) = (y; x) 8 x; y 2 E ;<br>then it is also called a hermitian sesquilinear form.<br>2. When K = R; that is, E is a real vector space, a hermitian (sesquilinear)<br>form on E is just called a symmetric bilinear form on E.<br>Denition 1.2.3<br>A linear space endowed with an inner product is called an inner product space<br>or a prehilbertian space.<br>A nite dimensional real prehilbertian space is also called a euclidean space<br>Theorem 1.2.4 (Cauchy-Schwarz-Bunyakovsky Inequal-<br>ity)<br>Let (E; h ; i) be an inner product space. Then<br>jhx; yij2 hx; xi hy; yi 8 x; y 2 E:<br>The equality of this inequality holds if and only if x and y are linearly dependent.<br>Theorem 1.2.5 (Norm induced by an inner product)<br>Let (E; h ; i) be an inner product space. Then the function<br>jj x jjE : E ô€€€! R<br>x 7ô€€€!<br>p<br>hx; xi<br>denes a norm on E:<br>Denition 1.2.6 (Hilbert space)<br>An inner product space E is called a Hilbert space (usually denoted by H), if<br>(E; jj : jjE) is a Banach space.<br>19<br>Remark 1.2.7 (Hilbert space)<br>Finite dimensional real Hilbert spaces are also called Euclidean spaces. And<br>since all nite dimensional normed spaces are Banach spaces, there is no dierence<br>between nite dimensional real prehilbertian spaces and nite dimensional<br>real Hilbert spaces.<br>Theorem 1.2.8 (Parallelogram law and Polarization identity)<br>Let (E; h ; i) be an inner product space. Then we have<br>1. the parallelogram law :<br>jjx + yjj2 + jjx ô€€€ yjj2 = 2(jjxjj2 + jjyjj2) 8 x; y 2 E ;<br>2. and the Polarization identity :<br>hx; yi =<br>1<br>4</p><p>jjx+yjj2ô€€€jjxô€€€yjj2+ijjx+iyjj2ô€€€ijjxô€€€iyjj2</p><p>8 x; y 2 E :<br>Note that the parallelogram law characterizes the norms that are induced<br>by inner products according to a theorem by Von Neumann.<br>Denitions 1.2.9 (Orthogonality)<br>Let (E; h ; i) be an inner product space.<br>Two vectors x and y in E are said to be orthogonal (written x?y and read x<br>‘perp’ y) if hx; yi = 0:<br>If M is a non empty subset of E, we write x?M (and read x orthogonal<br>to M) if x is orthogonal to every element of M.<br>Given a non-empty subset M of E, we denote by M?, the set of all elements<br>of E which are orthogonal to M: That is,<br>M? =<br>n<br>x 2 E ; hx; yi = 0 8 y 2 M<br>o<br>:<br>The set M? is then called the orthogonal of M, it is a closed vector subspace<br>of E.<br>Theorem 1.2.10 (Projection Theorem)<br>Let H be a Hilbert space and M a closed subspace of H. For arbitrary vector<br>x in H, there exists a unique vector y? 2 M such that,<br>jjx ô€€€ y?jjH = inf<br>y2M<br>jjx ô€€€ yjjH:<br>Furthermore, x 2 M is such a vector if and only if (x ô€€€ x)?M.<br>The Projection Theorem yields the following denition end theorems.<br>20<br>Denition 1.2.11 (Direct Sum of vectors spaces)<br>Let X and Y be two subspaces of a vector space E. Then E is said to be the<br>direct sum of X and Y if<br>E = X + Y and X Y = f0g .<br>This means that every vector u 2 E has a unique decomposition of the form<br>u = x + y with x 2 X and y 2 Y:<br>in this case we write E = X Y .<br>Theorem 1.2.12 (Direct Sum Decomposition)<br>Let F be a closed subspace of a Hilbert space H. Then,<br>H = F F? :<br>Therefore we say that F? is the orthogonal complement of F<br>Theorem 1.2.13 (Riesz Representation)<br>Let H be a Hilbert space and let f be a bounded linear functional on H. Then,<br>(i) There exists a unique vector yo in H such that<br>f(x) = hx; yoi; for every x 2 H:<br>(ii) Moreover, jjfjjH = jjyojjH:<br>Corollary 1.2.14<br>If H be a Hilbert space, then H ‘ H via the canonical map<br>‘ : H ô€€€! H<br>a 7ô€€€! ‘a ;<br>where ‘a is dened by<br>‘a = hx; ai ; 8 x 2 H :<br>This canonical map is bijective, isometric and antilinear.<br>Corollary 1.2.15<br>Real Euclidean and Hilbert spaces are re exive.<br>21<br>Denitions 1.2.16 (Hilbertian basis)<br>Let H be a Hilbert space.<br>1. A unit vector of H is a vector of H is equal to 1.<br>2. A family</p><p>u</p><p>2ô€€€ of nonzero vectors of H is said to be orthogonal, if the<br>vectors of the family are pairwise orthogonal; that is,<br>hu ; ui = 0 ; for all 6= in ô€€€ :<br>3. A family</p><p>u</p><p>2ô€€€ of (nonzero) vectors of H is said to be orthonormal<br>if these vectors are all unit vectors and pairwise orthogonal; that is,<br>8&lt;<br>:<br>jjujj =<br>p<br>hu ; ui = 1 ; for all 2 ô€€€:<br>hu ; ui = 0 ; for all 6= in ô€€€ :<br>4. A family A of vectors of H is said to be total or complete if the vector<br>subspace spanned by A is dense is H; that is,<br>Span(A ) = H :<br>5. A family of vectors of H that is both orthonormal and complete is called<br>a hilbertian basis.<br>A hilbertian basis can be nite, countable or uncountable.<br>The crucial dierence between a hilbertian basis and a (nite) orthonor-<br>mal basis is that in the rst case the expansion of a vector may not be a<br>linear combination of some of the elements of the hilbertian basis, but a<br>series of vectors!<br>Examples 1.2.17<br>In the real space<br>`2 =<br>(<br>u = (un)n1 R;<br>X1<br>n=1<br>u2<br>n &lt; 1<br>)<br>endowed with the inner product dened by<br>hu; vi =<br>X1<br>n=1<br>unvn ;<br>the following vectors orthonormal and complete.<br>e1 = (1; 0; 0; 0; : : <img alt="" src="https://s.w.org/images/core/emoji/11/svg/1f642.svg"><br>…<br>ek = (k;n)n where k;n = 1 si n = k and k;n = 0 otherwise<br>…<br>22<br>Proposition 1.2.18<br>Let H be a Hilbert space. Then<br>1. For every vector subspace F of H, we have<br>ô€€€<br>F??<br>= F :<br>In particular if F is a closed subspace, then<br>ô€€€<br>F?<br>?<br>= F :<br>2. For every nonempty subset A of H, A? is a closed vector subspace of H<br>(and so a Hilbert subspace) and<br>ô€€€<br>A??<br>= Span(A) :<br>3. A nonempty set S of H is complete (or total) if and only if<br>S? = f0g :<br>Theorem 1.2.19 [17]<br>Let (H; h ; i) be a Hilbert space. Then<br>1. H has a hilbertian basis (i.e., an orthonormal basis) that can be nite<br>countable or uncountable. Furthermore, every orthonormal set in H is<br>contained in some hermitian basis.<br>2. A hilbertian basis of H is at most countable if and only if H is separable;<br>that is, H contains a dense subset that is at most countable.<br>3. For every orthonormal sequence of vectors fekgk2N of elements of H, we<br>have Bessel Inequality:<br>8 x 2 H;<br>X+1<br>k=1<br>jhx; ekij2 jjxjj2 :<br>4. If H is separable and fekgk2N is a complete and orthonormal sequence<br>of vectors of H, then we have Parseval Identity:<br>8 x 2 H;<br>X+1<br>k=1<br>jhx; ekij2 = jjxjj2 :<br>23<br>Proposition 1.2.20<br>Let (H; h ; i) be an innite dimensional Hilbert space and fekgk2N be an<br>orthonormal set of H. Then<br>1. For every x 2 H, we have<br>lim<br>k!+1<br>hx; eki = 0 :<br>2. Every bounded sequence of scalars fkgk2N gives a bounded linear operator<br>T on H dened by<br>Tx =<br>X+1<br>k=1<br>khx; ekiek ; 8x 2 H ;<br>of which norm is jjTjj = supk1 jkj.<br>Proof<br>1. Follows from the convergence of the numerical series<br>P+1<br>k=1 jhx; ekij2<br>according to Bessel inequality.<br>2. Follows from the convergence in H of the series<br>P+1<br>k=1 khx; ekij2 and<br>the bounds<br>jjTxjj sup<br>k1<br>jkj jjxjj ; 8x 2 H<br>and<br>sup<br>k1<br>jjTekjj = sup<br>k1<br>jkj :<br>Theorem 1.2.21 (Construction of compact linear operators) [24]<br>Let (H; h ; i) be an innite dimensional, separable complex Hilbert space with<br>orthonormal basis (ek)k2N and let (k)k2N be an arbitrary sequence of<br>complex numbers. For every x 2 H, consider the series<br>Tx =<br>X+1<br>k=1<br>khx; ekiek :<br>Then<br>1. The series is convergent and the sum denes a linear operator T on H if<br>the sequence (k)k2N is bounded.<br>2. T exists and is bounded if and only if the sequence (k)k2N is bounded.<br>When k 2 f0; 1g for all k, T is an orthogonal projection.<br>3. T exists and is a compact linear operator if and only if the sequence<br>(k)k2N converges to 0.<br>When only nitely many of the terms k are nonzero, T is a nite rank<br>operator.<br>24<br>Denition 1.2.22 (Numerical range)[19]<br>Let H be a complex Hilbert space and T be a bounded linear operator on H.<br>The numerical range of T is dened by<br>W(T) =<br>n<br>hTx; xi : x 2 H; jjxjj = 1<br>o<br>:<br>Proposition 1.2.23 [19]<br>Let H be a complex Hilbert space and T be a bounded linear operator on H.<br>Then<br>(T) W(T) ;<br>where<br>W(T) =<br>n<br>hTx; xi : x 2 H; jjxjj = 1<br>o<br>: :<br>More precisely, if 62 W(T), then 2 (T) with</p><p>(T ô€€€ I)ô€€€1</p><p>1<br>dist(; W(T)<br>:<br>Denition 1.2.24 [6],[27]<br>Let H be a complex Hilbert space and T be a bounded linear operator on H.<br>The numerical radius of T is dened by<br>w(T) = sup</p><p>jhTx; xij : x 2 H; jjxjj = 1<br>o<br>:<br>It satises the following inequalities<br>r(T) w(T) and<br>jjTjj<br>2<br>w(T) jjTjj :<br>As a result of the Riesz representation Theorem, we have the following characterization<br>of weak convergence in a Hilbert space.<br>Proposition 1.2.25<br>Let H be a Hilbert space.<br>A sequence fxngn2N of elements of H converges weakly to some a 2 H if and<br>only if<br>lim<br>n!+1<br>hxn; yi = ha; yi ; 8 y 2 H :<br>25<br>Theorem 1.2.26 [24],[6]<br>Let H be a Hilbert space and (xn)n2N be a weakly convergent sequence with<br>limit point x. then for every compact linear operator T on H, the image<br>sequence<br>ô€€€<br>Txn</p><p>n2N converges strongly (in norm) to Tx.<br>That is, for any compact linear operator T 2 B(H), we have<br>xn * x for n ! +1 =) Txn ! Tx for n ! +1:<br>Corollary 1.2.27<br>Let (en)n2N be an orthonormal sequence of a Hilbert space H. Then for every<br>compact linear operator T of H, we have<br>lim<br>n!+1<br>Ten = 0 with respect to the norm topology of H :<br>Proof. By Bessel inequality, we have that (en)n2N converges weakly to 0 in H.<br>Therefore the corollary follows from the above Theorem [ ].<br>1.3 Self-adjoint operators<br>Let n be a natural number and T : Cn ! Cn be a linear operator. Then T<br>can be represented by a complex square matrix of order n. Suppose moreover<br>that T is self-adjoint. Then the matrix associated to T is conjugate symmetric<br>and it is well-known from basic Linear Algebra that all the eigenvalues of this<br>matrix are all real and that there exists an orthonormal basis (e1; : : : ; en) for<br>Cn in which the matrix of T is diagonal, meaning also that the linear operator<br>T can be expressed as follows :<br>Tx =<br>Xn<br>k=1<br>khx; ekiek ; 8 x 2 Cn ;<br>where hx; eki coincide with kth coordinate of x in the orthonormal basis (e1; : : : ; en)<br>of eigenvectors for T corresponding respectively to the eigenvalues</p><p>k</p><p>1kn.<br>For innite dimensional Hilbert spaces, the situation is much more tremendous,<br>but for self-adjoint compact linear operators, a corresponding theory can<br>be well developped. It will be culminated in the famous spectral theorem.<br>Denition 1.3.1 (Symmetric or self-adjoint operators) [19],[18],[20],[24]<br>Let H be a Hilbert space over K and A : H ! H be a bounded linear operator.<br>A is said to be symmetric or self-adjoint if<br>hAx; yi = hx; Ayi ; 8 x; y 2 H :<br>26<br>Remark 1.3.2 [24],[22]<br>1. Given an arbitrary bounded linear operator T on a Hilbert space H, the<br>adjoint operator of T is the unique bounded linear operator of H denoted<br>by T and satisfying<br>hTx; yi = hx; Tyi ; 8 x; y 2 H :<br>The existence and properties of T are based on the Riesz representation<br>theorem and can be proved following the idea of the proof of [Representation<br>theorem of a real bounded bilinear form].<br>Therefore a bounded linear operator T on a Hilbert space is symmetric<br>or self-adjoint if T = T.<br>2. Let H be a Hilbert space. Then any map T dened from H into H that<br>satises<br>hTx; yi = hx; Tyi ; 8 x; y 2 H ;<br>is necessarily linear and bounded. In fact :<br>a) the linearity holds since for all x; y 2 H and for all ; 2 K, we<br>have<br>T(x + y) ô€€€ Tx ô€€€ Ty</p><p>2 H? = f0g ;<br>b) and the boundedness follows from the Closed graph Theorem […]<br>3. Given a Hilbert space H and a linear operator T dened from a dense<br>domain D(T) H into H, the adjoint operator of T is dened as the<br>unique operator T with domain D(T) and such that<br>hTx; yi = hx; Tyi ; 8 x 2 D(T) and 8y 2 D(T) :<br>In this case, a linear unbounded operator T dened on a dense domain<br>D(T) of a Hilbert space H is said to be symmetric if T is an extension<br>of T, this amounts to:<br>hTx; yi = hx; Tyi ; 8 x; y 2 D(T) :<br>And again, a linear unbounded operator T dened on a dense domain<br>D(T) of a Hilbert space H is said to be self-adjoint if T = T. This<br>means that not only T is symmetric, but also D(T) = D(T).<br>Proposition 1.3.3 [24],[19]<br>The numerical range of any bounded, self-adjoint linear operator on a complex<br>Hilbert space is a subset of R.<br>In particular for any bounded, self-adjoint linear operator T on a Hilbert<br>space over K, hTx; xi is a real number for all x 2 H.<br>27<br>Proof. For all x 2 H, we have<br>hTx; xi = hx; Txi = hTx; xi :<br>Proposition 1.3.4 [24]<br>Let T be a bounded self-adjoint operator on a complex Hilbert space H. Then<br>all its eigenvalues are real numbers. Furthermore, any pair of eigenvectors<br>corresponding to dierent eigenvalues are orthogonal.<br>Proof. This is a well-known result.<br>If Tv = v for 2 C and v 2 H n f0g, we get<br>hv; vi = hv; vi = hTv; vi = hv; Tvi = hv; vi = hv; vi :<br>And since hv; vi = jjvjj2 6= 0; we get = meaning that is real.<br>Note that roughly prouving, is real because hTv; vi is real and hv; vi<br>is a nonzero real number.<br>If 1 6= 2 are two dierent eigenvalues corresponding respectively to two<br>eigenvectors v1 and v2, then 1 and 2 are real numbers and we have<br>1hv1; v2i = h1v1; v2i = hTv1; v2i<br>= hv1; Tv2i<br>= hv1; 2v2i = 2hv1; v2i = 2hv1; v2i :<br>Therefore<br>(1 ô€€€ 2)hv1; v2i = 0<br>and so hv1; v2i = 0 since 1 ô€€€ 2 6= 0.</p><p>We have the following alternative formular for the norm of a bounded se –<br>adjoint operator.<br>Proposition 1.3.5 [24],[19]<br>Let T be a bounded self-adjoint operator on a Hilbert space H (over K). Then<br>jjTjj = sup<br>x2H; jjxjj=1<br>jhTx; xij :<br>Proof.(Sketch)<br>Set = supjjxjj=1 jhTx; xij:<br>28<br>– First of all we have clearly jjTjj.<br>– Let x; y 2 H. By expending hT(x + y); x + yi and hT(x ô€€€ y); x ô€€€ yi<br>we get<br>hTx; yi + hTy; xi = 1<br>2</p><p>hT(x + y); x + yi ô€€€ hT(x ô€€€ y); x ô€€€ yi</p><p>2 (jjx + yjj2 + jjx ô€€€ yjj2)<br>(jjxjj2 + jjyjj2) (using parallelogram law):<br>And so<br>hTx; yi + hTy; xi<br>ô€€€<br>jjxjj2 + jjyjj2<br>:<br>– If x 2 H is such that Tx 6= 0, then by setting y = jjxjj<br>jjTxjjTx and by<br>applying the inequality of the previous step, we get<br>jjTxjj jjxjj :<br>The latter inequality is also (directly) satised even tough Tx = 0.<br>Therefore,<br>jjTxjj jjxjj ; 8x 2 H ;<br>yielding jjTjj .</p><p>Proposition 1.3.6 [18]<br>Let T be a bounded self-adjoint operator on a real Hilbert space H. Dene<br>the lower and upper bounds<br>m = inf<br>x2H; jjxjj=1<br>hTx; xi and M = sup<br>x2H; jjxjj=1<br>hTx; xi :<br>Then<br>1. (T) [m; M].<br>2. m; M 2 (T).<br>3. jjTjj = maxfô€€€m; Mg.<br>Corollary 1.3.7 [18]<br>1. For every bounded linear self-adjoint operator T on a real Hilbert space,<br>either jjTjj or ô€€€jjTjj is an approximate eigenvalue (i.e., a limit of a sequence<br>of eigenvalues of T).<br>2. Consequently, every nonzero compact linear self-adjoint operator T on a<br>real Hilbert space, has either jjTjj or ô€€€jjTjj as an eigenvalue; that is,</p><p>ô€€€ jjTjj; jjTjj</p><p>p(T) 6= ; :<br>29<br>1.4 Spectral decomposition<br>This section deals with the spectral decomposition of a compact linear selfadjoint<br>operator on a separable Hilbert space.<br>Theorem 1.4.1 (Hilbert-Schmidt) [24], [18], [28]<br>Let T be a compact self-adjoint operator on a separable Hilbert space H of<br>nite or innite dimension. Then H admits an at most countable orthonormal<br>basis consisting of eigenvectors for T. More precisely<br>1. In the nite dimensional case, the numbering of the nite sequence of<br>basis vectors (e1; :::; en) can be chosen such that the corresponding nite<br>sequence of eigenvalues (1; :::; n) decreases numerically (in absolute<br>value) :<br>j1j j2j : : : jnj :<br>And in the basis of eigenvectors, the operator T is described by :<br>Tx =<br>Xn<br>k=1<br>khx; ekiek for all x 2 H = Span</p><p>ek ; k = 1; :::; n</p><p>:<br>2. In the separable, innite dimensional case :<br>a) If T = 0, then any orthonormal basis of the separable Hilbert space<br>H is a countable orthonormal basis consisting of eigenvectors of T.<br>b) If T 6= 0, is of nite rank n 1, then<br>H = ker(T)R(T) with [ker(T)]? = R(T) and dim[R(T)] = n :<br>In this case R(T) is nite dimensional and invariant under T.<br>Therefore R(T) has a nite orthonomal basis (e1; :::; en) consisting<br>of eigenvectors of the restriction of T to its range R(T) such that<br>their corresponding eigenvalues satisfy<br>j1j j2j : : : jnj (all nonzero):<br>Again T is described by<br>Tx =<br>Xn<br>k=1<br>khx; ekiek for all x 2 H :<br>By adding to (e1; :::; en) an orthonormal basis of ker(T) (also separable),<br>we shall obtain a countable orthonormal basis for H consisting<br>of eigenvectors of T.<br>30<br>c) Otherwise<br>ô€€€<br>R(T) is innite dimensional</p><p>, we can nd an innite sequence<br>of orthonormal eigenvectors<br>ô€€€<br>ek</p><p>k2N ; in fact an orthonormal<br>basis of R(T) consisting of eigenvectors, with a corresponding sequence<br>of nonzero eigenvalues<br>ô€€€<br>k</p><p>k2N that decreases numerically<br>and tends to 0;<br>j1j j2j : : : jkj : : : with lim<br>k!+1<br>k = 0 ;<br>and for which T can be described as<br>Tx =<br>X+1<br>k=1<br>khx; ekiek for all x 2 H :<br>Note in this case that<br>p(T) f0g [</p><p>k : k = 1; 2; : : :</p><p>:<br>Proof (Sketch).<br>If the range of T is nite dimensional, then we can proceed by induction.<br>Now we only consider the case in which the range of T is innite dimensional.<br>– Set 0 = 0 and let<br>1 ; 2 ; : : :</p><p>be the countable set of all the nonzero eigenvalues of T. Consider the<br>subspace<br>H0 = ker(T) (that may be null) ;<br>and the eigenspaces<br>Hk = ker(T ô€€€ kI) ; k = 1; 2; : : : :<br>For k 1, dim(Hk) 1 since Hk is an eigenspace, and dim(Hk) &lt;<br>+1 since T is compact (cf. Theorem 1.1.48, by which every nonzero<br>eigenvalue k must have a nite multiplicity). Thus<br>1 dim(Hk) &lt; +1; 8 k 1 :<br>1. For any m 6= n in</p><p>0; 1; 2 : : : g, Hm and Hn are orthogonal. (See the<br>proof of theorem …)<br>2. We have H = k0Hk where (recall)<br>k0Hk =</p><p>vk0 + : : : + vkn : n 0; 0 j n; kj 0; vkj 2 Hkj</p><p>:<br>To see this, set<br>V = k=0Hk<br>31<br>and suppose by contradiction that V 6= H. Therefore V ? 6= f0g and<br>V ? ker(T) V ? H0 V ? V = f0g :<br>Moreover, it is not hard to show that T maps V ? into V ?. Thus the<br>restriction of T to V ? would be a nonzero, compact self-adjoint operator<br>of V ? and would have at least one nonzero eigenvalue. This would imply<br>by Corrolary 1.3.7 that T has an eigenvector in V ? with a nonzero<br>eigenvalue contradicting the fact that<br>V ? [k1Hk V ? V = f0g :<br>3. For each k 1, the nite dimensional subspace Hk possesses a nite<br>orthonormal basis<br>Bk =<br>n<br>ek; 1 ; ek; 2 ; : : : ; ek; nk<br>o<br>:<br>Besides the closed subspace H0 = ker(T) of the separable space H, is<br>either null, in which case we set B0 = ;, or admits an at most countable<br>orthonormal basis B0.<br>It follows that<br>B = [+1<br>k=0Bk<br>is a countable orthonormal basis of eigenvectors of T. Furthermore<br>Tx =<br>X+1<br>k=1<br>Xnk<br>j=1<br>khx; ek; jiek; j for all x 2 H :<br>Constructive proof (Sketchy)<br>We shall prove again this Theorem 1.4.1 by successive applications of Corollary<br>1.3.7.<br>Let H1 = H assumed to be nontrivial and set T1 = T.<br>By the second part of Corollary 1.3.7,<br>there exist an eigenvalue 1 of T1 and a corresponding eigenvector ‘1 such<br>that jj’1jj = 1 and j1j = jjT1jj. Set H2 := f’1g?. Thus H2 is a closed<br>subspace of H1 and T(H2) H2 (i.e H2 is T-invariant).<br>Now let T2 be the restriction of T to H2. Then T2 is compact self adjoint<br>operator in B(H2)<br>If T2 6= 0,then there exists an eigenvalue 2 of T2 and corresponding eigenvector<br>‘2 such that jj’2jj = 1 and j2j = jjT2jj jjT1jj = j1j<br>f’1; ‘2g is orthonormal.<br>H3 = f’1; ‘2g?<br>H3 is a closed subspace of H and TH3 H3<br>32<br>Letting T3 be the restriction of T to H3, we have that T3 is a compact selfadjoint<br>operator in B(H3). Continuing in this manner, the process stops when<br>Tn = 0 or else we get a sequence fng of eigenvalues of T and corresponding<br>orthonormal set f’1; ‘2; ‘3:::g of eigenvectors such that<br>jn+1j = jjTn+1jj jjTnjj = jnj n = 1; 2; 3::: (1.4.1)<br>Claim : If fng is an innite sequence, then n ô€€€! 0; n ô€€€! 1.<br>Proof of Claim. Suppose by contradiction , there exist &gt; 0 such that jnj<br>for all n 2 N<br>Hence for n 6= m, we have that,<br>jjT’n ô€€€ T’mjj2 = jj’n ô€€€ ‘mjj2 = 2<br>n + 2<br>m &gt; (1.4.2)<br>But this is impossible, since fT’ng has a convergent subsequence due to the<br>compactness of T. We therefore conclude that n ô€€€! 0; n ô€€€! 1.<br>Now, we prove the representation of T as asserted in the theorem.<br>Case I. Tn = 0 for some n<br>xn := x ô€€€<br>Xn<br>k=1<br>&lt; x; ‘k &gt; ‘k<br>It is evident that xn is orthogonal to ‘i for 1 i n<br>Therefore, xn 2 Hn<br>0 = Tnxn = Tx ô€€€ T(<br>Xn<br>k=1<br>hx; ‘ki’k)<br>) Tx =<br>Xn<br>k=1<br>khx; ‘ki’k<br>Case II. Tn 6= 0 for all n 2 N<br>jjTx ô€€€<br>Xn<br>k=1<br>khx; ‘ki’kjj = jjTnxnjj jjTnjj jjxnjj<br>= jnj jjxnjj<br>jnjjjxjj ô€€€! 0<br>) jjTx ô€€€<br>Xn<br>k=1<br>khx; ‘ki’kjj as ô€€€! 0; n ô€€€! 1.<br>Hence, Tx =<br>X1<br>k=1<br>khx; ‘ki’k :<br>33</p><p>&nbsp;</p> <br><p></p>