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1 Definition and Basic Properties 1 De¯nition and basic properties All linear spaces will be over the scalar ¯eld K = C oder R. De¯nition. A Fr¶echet space is a metrizable, complete locally convex vector space. We recall that a sequence (xn)n2N in a topological vector space is a Cauchy sequence i® for every neighborhood of zero U there is n0 so that for n; m ¸ n0 we have xn ¡ xm 2 U. Of course a metrizable topological vector space is complete i® every Cauchy sequence is convergent. If E is a vector space and A ½ E absolutely convex then the Minkowski functional k kA is de¯ned as kxkA = inff t > 0 j x 2 tA g: k kA is a extendedS real valued seminorm and kxkA < 1 if and only if x 2 spanfAg = t>0 tA. \ ker k kA = f x j kxkA = 0 g = tA t>0 is the largest linear space contained in A. We have f x j kxkA < 1 g ½ A ½ f x j kxkA · 1 g: If p is a seminorm, A = f x j p(x) · 1 g then k kA = p. We set ^ EA = ( span(A)= ker k kA; k kA ) ; ^ where " " denotes the completion. Notice that kx + ykA = kxkA for y 2 ker k kA. So k kA de¯nes a norm on span(A)= ker k kA. If E is a topological vector space and A has an interior point, then k kA is continuous. In this case 0 is an interior point of A and ± A = f x j kxkA < 1 g; A¹ = f x j kxkA · 1 g: 1 A space is called locally convex if it has a basis of absolutely convex neigh- borhoods of zero. Since for such a neighborhood of zero U, which may be assumed open, k kU is continuous and x 2 U i® kxkU < 1, we have for a generalized sequence (x¿ )¿2T that x¿ ¡! x i® p(x¿ ¡ x) ¡! 0 for every continuous seminorm. De¯nition. A set P of continuous seminorms on the locally convex space E is called fundamental system i® for every continuous seminorm q there is p 2 P and C > 0 so that q · C ¢ p. Of course, we have for every generalized sequence (x¿ )¿2T that x¿ ¡! x if and only if p(x¿ ¡ x) ¡! 0 for all p 2 P. If a fundamental system of seminorms P is countable then we may assume that P = f k kk j x 2 N g where k k1 · k k2 · :::. This can be achieved by setting kxkk = max pj(x) where f pj j j 2 N g is a given countable j=1;:::;k fundamental system of seminorms. Lemma 1.1 For a locally convex space E the following are equivalent: (1) E is metrizable. (2) E has a countable basis of neighborhoods of zero. (3) E has a countable fundamental system of seminorms. (4) The topology of E can be given by a translation invariant metric. Proof. (1) ) (2) and (4) ) (1) are obvious. To show (2) ) (3) we assume that U1 ⊃ U2 ⊃ ::: is a basis of absolutely convex neighborhoods of zero. We set k kk := k kUk . Then all k kk are continuous seminorms, then we may choose k so that Uk ½ f x j p(x) · 1 g. This implies p(x) · kxkk for all x. Finally, for (3) ) (4) we put X1 1 kx ¡ ykk d(x; y) = k 2 1 + kx ¡ ykk k=1 : 2 It is an elementary exercise to show that d(¢; ¢) is a translation invariant metric which gives the topology of E. 2 We notice that for a translation invariant metric d on a topological vector space E is complete if and only if E is complete with respect to d. For a not translation invariant metric this needs not to be the case as the example of R with the metric d(x; y) = k arctan x ¡ arctan yk shows. As in the case of Banach space we show: Theorem 1.2 If E is a Fr¶echetspace and F ½ E a closed subspace then F and E=F are Fr¶echetspaces. S A subset M of a linear space E is called absorbant if t>0 tM = E.A topological vector space E is called barrelled i® every closed, absolutely convex, absorbant set ("barrel ") is a neighborhood of zero. Theorem 1.3 Every Fr¶echetspace is barrelled. S Proof. Obviously E = n2N nM. By Baire's theorem there is n0 so that n0M and therefore also M has an interior point. So 0 is an interior point of M, which had to be proved. 2 Lemma 1.4 If E and F are locally convex, F a Fr¶echetspace and A: E ¡! F linear and surjective, then A is nearly open , i.e. for every neighborhood of zero U ½ E the set AU is a neighborhood of zero in F . Proof. We may assume that U is absolutely convex, hence AU is a barrel in F . Theorem 1.3 yields the result. 2 As in the case of Banach spaces we prove the following lemma. Lemma 1.5 (Schauder) If E and F are metrizable spaces, E complete, and A: E ¡! F linear, continuous and nearly open, then A is surjective and open. 3 These two lemmas yield: Theorem 1.6 (Open Mapping Theorem) If E and F are Fr¶echetspaces, A: E ¡! F linear, continuous and surjective, then A is open. Corollary 1.7 (Banach's Isomorphy Theorem) If E and F are Fr¶echet spaces, A: E ¡! F linear, continuous and bijective, then A¡1 is continuous. Theorem 1.8 (Closed Graph Theorem) If E and F are Fr¶echetspaces, A: E ¡! F linear and Graph A = f (x; Ax) j x 2 E g closed in E £ F , then A is continuous. Proof. We consider the maps ¼1 :(x; y) 7! x, ¼2 :(x; y) 7! y from Graph A to E, resp. F , we notice that ¼1 is continuous linear and bijective, hence by ¡1 ¡1 Corollary 1.7 ¼1 is continuous and therefore also A = ¼2 ± ¼1 . 2 If E is a Fr¶echet space, k k1 · k k2 · ::: a fundamental system of seminorms, ^ then the Banach spaces Ek := (E=k kk; k kk) are called "local Banach spaces ". Since ker k kk+1 ½ ker k kk we have natural quotient maps E=ker k kk+1 ¡! k E=ker k kk, which extend to continuous linear maps {k+1 : Ek+1 ¡! Ek, n ("linking maps "). Of course we may de¯ne in a similar way {m : Em ¡! En n n m¡1 k for m ¸ n and obtain {m = {n+1 ± ::: ± {m . By { : E ¡! Ek we denote the n m n quotient map. We have for m > n that {m ± { = { . We de¯ne continuous linear maps k {: E ¡! ¦kEk by {x = ({ x)k2N k ¾ : ¦kEk ¡! ¦kEk by ¾((xk)k2N) = (xk ¡ {k+1xk+1)k2N: Theorem 1.9 By { ¾ 0 ¡¡¡¡! E ¡¡¡¡! ¦kEk ¡¡¡¡! ¦kEk ¡¡¡¡! 0 we obtain a short exact sequence ("canonical resolution"). n Proof. { is clearly injective since { = 0 for all k implies kxkk = 0 for all k, hence x = 0. 4 k k k+1 k k im { ½ ker ¾ because { x ¡ {k+1{ x = { x ¡ { x = 0. To prove ker ¾ ½ im ¾ we notice that k ker ¾ = f (xk)k2N j xk = {k+1xk+1 for all k g: n This implies xn = {mxm for all n · m. Let x 2 ker ¾. For every xk we k ¡k choose »k 2 E so that kxk ¡ { »kkk · 2 . For k · n < m we obtain k k k k k»n ¡ »mkk = k{ »n ¡ {nxn + {mxm ¡ { »mkk n m · k{ »n ¡ xnkn + kxm ¡ { »mkm · 2¡k+1 Therefore (»n)n is a Cauchy sequence, hence convergent. We put x = limk!1 »k. We have k k kxk ¡ { xkk = lim kxk ¡ { »nkk n!1 k n n = lim k{ xn ¡ { { »nkk n!1 n n n · lim kxn ¡ { »nkn = 0: n!1 k So xk = { x for all k. To prove surjectivity of ¾ we consider an element (0;:::; 0; xk; 0 ::: ), xk 2 k ¡k Ek. There exists »k 2 E such that kxk ¡ { »kkk < 2 . We set yk = k ¡k xk ¡ { »k 2 Ek, then kykkk < 2 . We de¯ne elements k+1 k+2 ak = (0;:::; 0; ¡{ »k; ¡{ »k;::: ) 1 k¡1 bk = ({kyk;:::;{k yk; yk; 0;::: ): Then we have k ¾(ak) = (0;:::; 0{ »k; 0 ::: ) ¾(bk) = (0;:::; 0; yk; 0;::: ): k k Now we put uk = ak+bk =: (´1 ; ´2 ;::: ). We obtain ¾(uk) = (0;:::; 0; xk; 0 ::: ) k ¡j with Pk´j kj < 2 for jQ= 1; : : : ; k. From thereP it is easily seen that u := k uk converges in k Ek. We have ¾(u) = k ¾(uk) = x. 2 5 2 The dual space of a Fr¶echet space For any locally convex space E we need the following notation for polar sets: For M ½ E we put M ± = f y 2 E0 j jy(x)j · 1 for all x 2 M g and for N ½ E0 we put N ± = f x 2 E j jy(x)j · 1 for all y 2 N g: If E0 is equipped with a locally convex topology ( e.g. the strong topology, see below ) then the second notation might be ambiguous. If there is any danger of confusion we will specify whether we mean N ± ½ E or N ± ½ E00. We recall Theorem 2.1 (Bipolar Theorem) If E is locally convex and M ½ E absolutely convex then M ± ± = M. If p is a continuous seminorm on the locally convex space E and U = f x 2 E j p(x) · 1 g then we have U ± = f y 2 E0 j sup jy(x)j · 1 g: x2U We put p¤(y) = sup jy(x)j x2U and obtain U ± = f y j p¤(y) · 1 g: Therefore p¤ is the Minkowski functional of U ±.
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