PDF hosted at the Radboud Repository of the Radboud University Nijmegen The following full text is a publisher's version. For additional information about this publication click this link. http://hdl.handle.net/2066/60069 Please be advised that this information was generated on 2015-02-05 and may be subject to change. ORTHOCOMPLEMENTATION IN p-ADIC BANACH SPACES C . PereZ'Garcia* and W . H . S c h i k h o f A bstract. This paper deaJs with classes of Banach spaces E over an o n -arch im ed ean valued field for which every one-dimensional subspace satisfies some orthocom plem entation property. They are described in term s of polarity for the balls of E (section 1) and in term s of compactoidity for the balls of the dual space E' (section 2). This stu d y yields (L em m a 1.6) the solution of an open problem raised by the second a u th o r in [8]. Finally, the stability properties of these spaces are discussed in section 3. 0. P R E L I M I N A R I E S Throughout K is a non-archim edean valued field th a t is complete with respect to the metric induced by the non-trivial valuation | • |, By |Ar| we will denote the set {|A| : A € A'}. For fundam entals on Danach spaccs and locally convex spaces over K we refer to [9] and [4] respectively. Let £ be a K - vector space. A subset A of E is absolutely convex if it is a m odule over the ring (À e K : |A| < 1}. For a set D in E we denote by [B] th e A"-vector space generated by 5 , and by coZ? the sm allest absolutely convex set of E containing B. For an absolutely convex set A in E we set Ae := D{Aw4 : |A| > 1} if the valuation of K is dense, A e := A otherwise, and .4' := U{AA : |A| < 1). For a (non-archim edean) scminorm p on E , Kerp will be the set {.t G E : p(x) = 0}. Recall th at p is called polar if p = sup {Iƒ I : ƒ 6 £ p }, where E* is the algebraic dual space of E . Let E be a locally convex space over K . For a set B in E we denote by [Ö] the closed linear hull of B . An absolutely convex set A in £ is called: a) compactoid: if for each neighbourhood U of 0 there exist n 6 N a n d x \ , . . . , € E such that A C U + c o ) i ] , . . . , i n ), b) c'-compact: if in the above we m ay choose x j , . . . , a:„ (E A, * Research partially supported by Comision M ixta C a ja C a n ta b ria - Universidad de Cantabria 35 c) compactoid of finite type: if for each neighbourhood U of 0 there exists a bounded finite-dimensional set 5 C A such th a t A C U + S. Clearly A c'-compact =>• A compactoid of finite type => converses are n o t true in general (see [5] and [7]). A compactoid. T he Let (E , Ij ■II) be a Banach space over K . For each r > 0, B&(r ) (resp. B e ( t )) will m ean {x Ç. E : ||a;i| < r} (resp. {x G E : ||x|| < r}). By E* we denote the topological d u al space of E. Recall that E* is again a Banach space endowed with the norm ||/* = s u p { l ^ M : * 6 B , : r ? 4 0 } (ƒ€£')• Let t 6 (0,1] and let S ,T be sets in E . We say th at 5 is f-orthogonal to T (and write S J-tT ) if for all x 6 5, y 6 T: IlAæ + fiy\\ > t max(||Ax||, ||/iy||) (A,/i 6 K ) . If I is an index set, a family {e* : i 6 1} of elements of E is called a base of E if for each x £ E there is a unique (Aj){g/ £ I \ ! such th a t x = If in addition II E f e J ~ imSiXitj ||or,a:,-1| for all 3 C / , J finite and all a* Ç K (i 6 J ), we say th a t {e* : i 6 1} is a t -orthogonal base of E. For t — 1, we write orthogonal instead of 1-orthogonal and _L instead of _Li. Now, let D be a finite-dimensional subspace of (E , || * ||). We say that D has an orthogonal almost complement if there exists a closed subspace H of finite codimension in E such th a t D l H , Such an H is called an orthogonal almost complement of D. (If, in addition, D + H s= E it is custom ary to drop the word ‘alm o st’ in the above.) Also, D is called almost orthocomplem ented if for each t G (0,1) there exists a closed subspace H t of finite codimension in E such th a t D X t E i and D + Hi = E. Such an H t is called a ^-orthogonal complement of D. Recall th a t ( E %|| • ||) is called norm-polax if || • || is a polar norm on E (or equivalently, every one-dimensional subspace of E is almost orthocom plem ented, Theo rem 1.2). We also consider in this paper the following related classes of Banach spaces: E is called Hilbertian (resp. almost H ilbertian) if every one-dimensional subspace of E is orthocom plem ented (resp. has an orthogonal almost complement). One verifies (for the second implication see Proposition 3.5(ii)) E Hilbertian E almost H ilbertian =» E norm -polar. Also, if K is spherically complete, then every Banach space over K is Hilbertian ([9], Lem m a 4.35). Hence, FROM NOW ON IN THIS P A P E R E WILL BE A BANACH SPACE OVER A NON-SPHERICALLY C O M P L E T E FIELD K . 36 1. H I L B E R T I A N S P A C E S A N D P O L A R S E T S For an absolutely convex set A C E we set (see [4]) A 0 = {ƒ £ E' : |/(ci)| < 1 for all a € A} A °°= {x £ E : |/( .r ) | < 1 for all ƒ € A 0} A is called a polar set if A = A°°. In the same spirit we define (see [10]) A ° = { ƒ € £ ' : |/ ( a ) | < 1 for all a Ç A } ADD= Iƒ (a:) I < 1 for all ƒ e AD} A is called a pseudopolar set if A = Aa D . One verifies that .400 = ( A^) e ([4], Prop. 4.10.) and th at A C Aa C ADD C A°° (/) for every absolutely set A C E (where A ° denotes the closure of A with respect to the weak topology < j (B, E') on E ). We construct a set A for which all the inclusions appearing in (I) arc strict. E x a m p le 1.1. Let E = A" ® A'J 0 J*1, where Ar;J is the two-dimensional Danacli space constructed in [9], p. 68, and where F is the vector space Co endowed with a norm N which is equivalent to the suprcm um norm on c0, but with the property t h a t if x , y £ F are such t hat x is IV-orthogonal to y, then x* = y == 0 (to see th a t such a norm exists, consider in the spherical completion K of A", a sequence a \ , a 2 , . . ■consisting of A'-linearly independent elements. By taking 011 th e norm induced by the valuation of A", we obtain an infinite-dimensional Banach space of countable type over K y for which there are no non-zero m utually orthogonal elements, see [9), Exam ple 5.E). Let A C E be given by A = i? (£) T 0 S where /£, T and S are the open unit balls of A , A’j5 and F respectively. It is easy to see that _ A° = R Q )T ® Sc ADD= i ? e r e ® 5 c A°° = R e $ T c 0 S e and so A ^ A* ^ ADD ^ A°°. It is well-known (see [4]) th a t the norm -polar Banach spaces are precisely those Banach spaces E for which B ß { \ ) is polar. Norm-polar spaces can also be described in terms of a complem entation property or a Hahn-Banach property as follows, 37 Theorem 1.2., The following are equivalent i) E is norm-polar. ii) For each one-(finite-)dimensional subspace D } fo r each e > 0, and f o r each ƒ Ç D 1, there exists an extension f £ E ' such that j|/|| < (1 + e ) ||/ j |. iii) Every one-(finite-)diinensional subspace of E is almost orthocomplemented, iv) B&( 1) is polar. P r o o f . W ith a simple adaption of the proof of Lemma 4.35.iii) of [9] we can derive th a t if the one-dimensional version of iii) holds then so does the finite-dimensional one. So, it is enough to prove the theorem for the case of one-dimensional subspaces. The equivalences i) <=> ii) and i) <*=> iv) were proved in [1], Theorem 2.1 and [4], Proposition 3.4 respectively. ii) =*> iii): Let D = [#] (x* € E — {0}) be a one-dimensional subspace of E , let t £ (0,1). The linear map [.r] —►K : Ax —+ A has norm ||.r||- 1 . By ii), there exists an M P M * » . . ______ « v H extension ƒ £ E* such that ||/|| < . Then, K e r f is a i-orthogonal complement of D. iii) =*> ii): Let D be a one-dimensional subspace of let e > 0 and let ƒ £ D*. By iii), there exists a continuous linear projection P : E —* D with ||P || < 1 -}- e. Then, ƒ := ƒ o P £ E l extends ƒ and ||/|| < (1 e )||/||, R e m a r k 1.3, By using Propositions 3.4 and 4.10 of [4], it is not hard to see th a t iv) of above is also equivalent to each one of the following statem ents, v) B E(r is polar for each r > 0. vi) # e (1 is pseudopolar, vii) B B(r is pseudopolar for each r > 0, viii) B e { 1 is weakly closed. ix) B e ( t is weakly closed for each r > 0. Now we are going to describe the Banach spaces E for which B $ ( r ) is pseudopolar (weakly closed) for each r > 0. In the same vein as Theorem 1.2 we can prove: T h e o r e m 1.4. The following are equivalent, i) E is Hilbertian. ii) For each one-(finite-)dimensional subspace D, for each f £ D ' , there exists an extension f £ E ' such that ||/ || = iii) Every finite-dimensional subspace of E is ortho complemented. iv) 2?ß(r~) is pseudopolar fo r each r > 0. P r o o f . The equivalence i) iii) was proved in [9], Lemma 4,35. Also, i) «*=> ii) follows in a similar way as ii) <=> iii) in Theorem 1.2, 38 ii) =£■ iv): Let r E (0, oo). Take a* G E , ||j:|| > r. By ii), there exists au f € E ' such th at | / ( i ) | = 1 and ||/|| = \\x\\~l (and hence ƒ G (i? E (r~ ))D). So, i £ (-Öe(7, _ ))DDiv) => i): Let x Ç. E {0} and let r := ||x||. By iv), there exists a g G E* such th at ||^ H < r “ 1 and | ^( t ) | > 1. Hence, |<7(a:)| = ||^|)||^||, which implies that Kcvg is an orthogonal complement of [a;] in E. We shall describe almost H ilbertian spaces in a similar way as we did in Theorems 1.2 and 1.4 for norm -polar and Hilbertian spaces respectively, To this end we need the following lemmas. L e m m a 1.5, Let D\ W\, be subspaccs of E, I f J-D 2, W 1 X W 2 a n d W ^ C D i, then D\ -}- W 2‘ ^ ' ^ 2 H W j. P r o o f , Let a G D \ ,6 G Wa, c G D 2 fl W j. Then 6 + c G D 2 and so ||a -f 6 + c|| > ||6 + c||. By Lemma 3.2 of [9] we obtain that ||a + 6 + c|| > ||a||. B ut also fcXc so that ||6 + c|| > max(||6||, ||c||). Hence, « + 6 + c\\ > max{ ||«||, ||i||, ||c||) > m ax(||a + 6||, ||c and the result follows. Lem m a 1,6, (This gives an affirmative answer to the problem raised in [8], §5.) If E is almost Hilbertian, then every finite-dimensional subspace of E has an orthogonal almost complement. ♦ Proof, It suffices to prove: if D is a finite-dimensional subspace of E having an orthogonal almost complement and if « G E — D , then D\ = D 4* [a] also has an orthogonal almost complement. In fact, there is a closed subspace H of E of finite codimension such that D ± H . Now we distinguish two cases. a) D is not orthocomplemented in D\, We prove that D\J l H. Let x G D\ J i G H. To see th a t z l / i we may assume a: £ D. Then x is not orthogonal to D so there is a d G D with ||a:—f/|| < ||ri|| = ||.r||. Then ||a; —/i|| = \\x —d + d —h\\ = max(||ar—c/||, 1)^—/i||) > d\\ = ||a*||. It follows easily t hat \\x — /?|| > max(||a:||, ||/i||) (Lemma 3.2 of [9]). b) D has an orthogonal complement in D \. We may assume th at aJLD. Also, we may assume th a t \a\ and D + H are not orthogonal (if they are then H X D \ ) y so there exists a v G D + H for which \\a — u|| < ||a||. Then aJLD implies u_LD, Write V — d + h (d G J9, h G H ). Since E is almost Hilbertian, there is a closed subspace S of E of finite codimension such th a t [/i]JLS. By Lemm a 1.5, D + I \ v = D - \- K h ± H C \ S . We finish the proof by showing th at D \ X H H S (observe that H f l S has finite codimension in E). For th at, let i Ç Û , A G K — {0}, c £ H D S. Since H a:+Av+c| >|Au|>|A||a- v 39 we deduce that \\x + A« + c|| = ||x + Xv + c|| > raax(||a:||, ||c||, ||Av||) = tnax(||x||,||c||,||Aa||) > > \\x 4* Aa|| and we are done. T h e o r e m 1*7* The following are equivalent i) E is almost Hilbertian, ii) For each one-(finite-)dimensional subspace D there exists a closed finite- codimensional subspace H of E such that D C H and every ƒ G D ’ admits an extension to an element of H* with the same norm. iii) Every finite-dimensional subspace of E has an orthogonal almost complement iv) B e {t ~) is weakly closed fo r each r > 0. P r o o f , The equivalence i) <=> iii) follows from Lemma 1.6. Also, i) ii) can be proved as ii) iii) in Theorem 1.2. i) =»> jv): Let r G (0, oo). Let x G E % ||x|| > r. By i), there is a closed subspace H of E with finite codimension in E such th at Let n : E —* E / H be the canonical surjection and let q be the norm on E / H associated to || • ||. Then, p — q o tt is a weakly continuous seminorm on E for which p(x) — ||z|| > r and p ( B E ( r ~ ) ) C [0, r ). .. a So, {y 6 E : p(ar) < r} is a weakly open set which contains B e { t ~ ) and which does not contain x. Hence, B e ( ? ~ ) is weakly closed, iv) =>■ i): Let x £ E — {0} and let v ||æ||. By iv), there is a weak neighbourhood U of 0 such that (x -f U ) D B e {v ~) = 0 i.e., ||arH-1/|| > ||x|| for all u G U. Now U contains a closed subspace H of finite codimension and since ||x /i|| > ||x|| for all h G H we have [ x j l t f , R e m a r k s 1 *8 . 1) In contrast to the statem ents appearing in Remark 1.3, there are Banach spaces E for which property iv) of Theorem 1,4 (resp. Theorem 1.7) is not equivalent to “2?e(1” ) is pseudopolar” (resp. uB e ( 1*") is weakly closed1'). E x a m p le . Suppose that |Ar| ^ [0,oo) and let r > 0 such th at r £ |Ar|, Let N be the norm on cq considered in Example 1.1 and let E := (cq,.s) where s(x) = r N ( x ) (x € E ). Clearly, j i 6 £ : 3 ( 1 ) < 1} = {a: G E : ,s(x) < 1 } 40 (recall th a t N { x ) 6 \K\ for all i Ç co). Hence, B e { 1” ) is polar (and hence pseudopolar and weakly closed). But there are not non-trivial mutually s -orthogonal elements ni E. So, E is not almost Hilbertian (and hence E is not Hilbertian). 2) Looking at properties iii) of Theorems 1.2, 1.4 and 1.7, it seems natural to consider the class of Banach spaces E satisfying the following property. “For each one-(finite-)dimensional subspace Z?, there exists a closed subspace H of finite codimension such th a t D is almost orthocomplemented in H % \ Since every closed subspace of finite codimension of a Banach space E is almost ortho complemented in E ([1], proof of Theorem 4.7.i) => ii)), we conclude that the above property is nothing b u t norm -polarity of E. 3) Also, looking at properties iv) of Theorems 1.2, 1.4 and 1.7 one might think of the following property for a Banach space E. llB E ( r - ) is polar for each r > 0” . But one can easily see that if E has this property then B g ( r ~ ) = i?£;(r) for each r > 0, and so E — {0}. 2. H I L B E R T I A N S P A C E S A N D C O M P A C T O I D S E T S In this section we give several new descriptions of Hilbertian and almost Hilbertian spaces in terms of compactoidity properties of the balls in the dual space. T he following lem ma will be crucial for our purpose. Lem m a 2.1. Let D be a finite"dimensional subspace of E . L e t n € N (n > 1). Consider the following statements, i) There exists a closed subspace. H of E with dim E / H — n such that D X H . Ä ii) For each r > 0 there exists 5 V C B[,j»(r~) vnth dim[Sr] = n such that B w ( r ~ ) C D° + Sr. iii) There exists S C i ? £ ' ( l - ) with diinfS] = n such that B[r>( 1“ ) C 4* S. Then we have i) =3* ii) ==> iii). I f in addition E is norm-polar, then i) - iii) arc equivalent. Proof, i) ^ ii): The formula q(x) = dist(.r, H ) = inf{||x — h\\ : h 6 H } defines a continuous semi norm on E with dim ( £ /Kerry) = n such that q < || * || and q — || • || on D. Now, let r > 0 be given. Let Tr = { f S E* : \ f \ < r q } . 41 1 We see th at dim[Tr] = n. Also, since q < || • || we have th at Tr C B g f ( r ) . We now shall prove th at % (r')C D ° + S r w here S r = (Tr )*. In fact, let ƒ G B&' ( r _ ). Then there is a A Ç / i , 0 < |A| < 1 with |ƒ j < |A |r|| • ||. Choose À' G K with |A| < |A'| < 1. Since q = || • || on D we have | / | < \X\rq on D y so we can extend ƒ to a g G E f with |<j| < |A'|r<y. (This is because d im (E /K e r q ) < oo so q is polar.) Now write / = (ƒ ~ g ) + gSince ƒ = g on D wo have ƒ —g £ D°. Also, |</| < |A*|rg and so (A') ~ 1g 6 Tr , which implies th a t </ G (Tr ) \ Clearly ii) =*> iii). Now, assume th at E is norm-polar. iii) =*> i): There exists S C B& ( 1"*) with dim[5] = n such th at = (D° n + 5. Set q(x) = sup/<e5 |/i(x)| (x G E). Then q is a continuous semi norm on E for which dim( E /K en /) = n, Also, since E is norm-polar, for each x G E we have q(x) < sup |/i(.t)| = sup |/i(x)| = ]|x|| It*l1<l I(A|I<1 a n d so q < || • ||. Further, if x G D then ||x|| = sup | / ( s ) | = sup{|/*.(x) + f(x)| : h G D° fl I?£y(l- ), t G S} = ll/ll <i = s u p |^ .r ) | = ÿ{.r) res and hence q = || • || on D. Finally, for d G D and y G Kerq we have: \\d + y\\ > q(d + y) = q(d) ~ \\d\\ which means th a t DJLKcrg. We conclude th at H = Kerg satisfies the conditions required in i). R em ark 2*2. Properties i) - iii) of Lemma 2.1 are not equivalent in general. E x a m p l e . Take E — (i°°yu) where v(x) = m ax(j|x||00,2dist(x,co)) 42 (x G C°°). Set n = 1 and D = [e] where e = ( 1 , 1 , 1 , . , . ) . Since D is orthocoinplem ented in (^°°, (I • IIoo) and the* identity m ap E 1 —+ (f°°,|| • ||oo)/ is an isom etry ([9], Ex.4I<) we deduce th at iii) is true for E. Now, we prove th at i) is not true for E . If it were true then D is orfchocomplemented in E and so there would be an ƒ G E ' - {0} such that |/( e ) | = j)/||||e||. Set ( a j , a 2, . .. ) G Co such th at f ( x ) = anx n for all z — ( i ! , x 2, *• 2m ax^Lj |a„|, which is a contradiction. G E. T h en , | a n\ = Next, we shall apply Lem m a 2.1 to characterize Hilbertian spaces by m eans the following variant of compactoidity: D e fin itio n 2.3. Let F be a locally convex space over K a n d let A be an absolutely convex subset of F. A is called nearly c/-compact if for every zero-neighbourhood U in F there exist n 6 N mad bounded sets S i , . . . , S n contained in A with dim[Xj] = 1 for al i i = 1 , . . . , n and such th a t A C U -f Si + . . . + S„. Observe th at A c'-compact A nearly c'~compact => => A compactoid of finite type. But the converses are not true in general (see Theorems 2,5 an d 2.6). Lemma 2*4. Lei F and A be as in Dcfinitioji 2,3. Suppose that the topology o f F is generated by a family V of non-archimedean seminorms such that d i m ( E / K e r p ) < oo fo r all p Ç;V. Then the following properties are equivalent, i) A is nearly c*-compact (resp. A is of finite type). ii) For every closed subspace H of finite codimension there are bounded sets S i , . . , , S n contained in A vnth diin[S,] = 1 for all i = l , . . . , n (resp, there exists a fin ite • dimensional bounded sei S C A ) such that A C H + S\ -j- . . . -f S„ (resp. A C H -f S). P r o o f . We prove the result for nearly r*-compact sets. For the case of sets of finite type the proof is similar. i) => ii) (Observe that this implication holds for any locally convex space F ) \ We may assume th a t [A] — F. H has the form H = D° where D is a finite-dimensional subspace of F*. Let l>e a base of D. There exist G F with f x( x j ) = Sij ( i , j = l,...,m). Since [A] = F , there exists a A G A' A ^ 0 such th at A.t,* G A for each i G { 1 , . . . , m) . Set m u = f ) { x e F : \f,{x)\ < \ \ \ ) . 1= 1 43 T h e n U is a zero-neighbourhood in F . Since A is nearly c'-compact there arc bounded subsets S \ , . . . , S r of A with dim[S/,] = 1 for all h = 1 , . . . , r such th a t A C U 4- S] 4.. . 4- S r . Let x G U . Write x — y 4- z where ?7l y ~ x - 5^/«(^)® « t-i m z = Y ji(x)X i. i=l Now, since a: Ç [/, we have that z G T\ 4* . . . -j- Tm where T* = co{Axj) (i = 1, *. . , m) (observe th a t T, C A for all t). Also, for j G { l , . . . , m } , f j ( y ) — 0 and so y G D°. Hence, we conclude th at A C H 4- T\ 4- . . + T1n 4- Si 4- . . . 4- S r a n d we are done. ii) i): Observe th a t every zero-neighbourhood in E contains a closed subspace of finite codimension. P u ttin g Lemmas 2.1 and 2.4 together we can now prove: T h e o r e m 2.5. The following statements i), ii) and iii) are equivalent, i) E is Hilbertian. it) E is norm-polar and B e >{ 1) is nearly c1-compact in a ( E \ E ) . iii) E is norm-polar and B £ / ( l ~ ) is nearly d-compact in a[E*yE). Also} the following statements i ’) and i i ’) are equivalent, i ’) E is Hilbertian and ||æ|| G |JV"| fo r all x G E. i i ’) E is norm-polar and B $t( 1) is cf -compact in c r ( E \ E ) . P ro o f. i) => ii): Clearly E is norm -polar (see Proposition 3.5). Now, let H be h a ( E \ £)-closed subspace of E 1 with dini(E ' / H ) = n < oo (n G N). T h e re are X\ ^. . . , ,r„ G E such th a t H = {ƒ G E f : ƒ ( .t,’) = 0 for all i — 1 , . . . , n }. For each m — 1 , . . . , u, set H ni = {ƒ G E 1 : f ( x { ) = 0 for all * = 1 , , m}. Since E is H ilbertian, it follows from Lemma 2.1 th at there exist a g G i ?£»( l ~) a n d an .s > 0 such th at £ ß< ( l - ) C t f , + S , 44 where Si = Bis {s) • (j- Clearly y % H i and so H \ D Si = {0}, which implies th a t Hence, T\ = (Sj )r is an absolutely convex subset of Æjj'O) with dim[Tj] = 1 such th a t B e ’( I ) C H\ 4- Ti (//) £e'(1) = #ƒ/,(!) + r, (iii) or, equivalently, On the other luuid, if M i is an orthogonal complement of [xj], we have that Hi is isometrically isomorphic to M[ via the isomorphism ƒ 6 Hi — ►ƒ \Mi € M |. T h is isometry maps H i onto a g )-closed subspace of M[. Since Mi is again Hilber tian, we can apply (II) to M \ instead of E to find a set T 2 C B n x( l) wi t h d i i n ^ ] = 1 such th at B h 1( \ ) C H 2 + T2 and by (III) it follows th at C H i + T\ + T2 Inductively, we can prove th at there exist subsets T j , . . . , T n of B & ( 1) witli dim[Tj] = 1 for all i — such th a t B ß ‘(l) C H *f T\ + . . . + T tl. Now, the nearly c'-compactness of B e *( 1) follows from Lemma 2.4. ii) =>• iii): One can easily prove th at if >1 is a nearly c'-compact subset of a locally » convex space, then A* is also nearly c'-compact. iii) =>■ i): Let D be a one-dimensional subspace of E, By iii) and Lem m a 2.4 there are absolutely convex sets £ * , , , , , S,f in Z?#<(1'") with dim[5,j = 1 for all i = l t . , , , n such th a t Z?e»(1” ) C D ü + S\ + . . . + S n . Also, since d im ( £ '/- ^ U) = * there exists m e { l , . . . , n } such th at n(Si ) 4* . . . 4- n ( S n ) = ?r(5m) (where ?r : E* —►£ ' / D i] is the canonical surjection) and so D 0 + Si 4-.. ■4- S n = D° + S m. Hence, B & i 1 ) C D° 4- S Vi which implies th a t D is orthocoinplemented (Lemma 2.1). i’) ii’): By [C], Theorem 3.2 it suffices to prove that m ax{|/(a;)| : ||/ || < 1} exists for each x 6 E, Since ||;i:|| 6 |A'| we may assume that ||æ|| = 1. For such x we m ust prove inax{|/(*‘)| • ||/ || < ! } = !< Since E is Hilbertian, [;r] has an orthogonal complement H . For the function ƒ : Xx -1- h —* À ( A G A', h G H ) we have |/ ( x ) | = 1. Also, for A E K , h e H t |/(A.r + h)\ = IAI = 11A*t 11 < max(||Ao:||, \\h\\) = || Ax 4- h 45 so th at U/H < 1. ii’) i’): It is straightforw ard to verify th at if A is a c'-compact set of a locally convex space, then A 1 is nearly c'-compact. Hence, ii’) implies iii), which is i). Further, by norm-polarity and c'-compactness, for each x G E we have ||x|| = su p {|ƒ (a:) | : ƒ G # £ '( ! ) } = m a x { |/(x )| : / € B e '( 1)} € |A'|. Also, as a direct consequence of Lemmas 2.1 and 2.4 we derive: T h e o r e m 2.6. The following are equivalent, i) E is almost Hilbertian. ii) E is norm-polar and, B e *{l - ) is of finite type in a ^ E ^ E ) . R e m a r k s 2.7. 1) Observe that with the same proof as in Theorem 2.5 we can see th at E is Hilbertian if and only if E satisfies one of the two following equivalent conditions: ii1) E is norm-polar and B e >(t ) is nearly c'-compact in a ( E \ E ) for each t* > 0. iii’) E is norm-polar a n d B e '( t ~) is nearly c'-compact in a ( E \ E ) for each r > 0. Analogously, one verifies th at E is almost Hilbertian if and only if E satisfies ii’) E is norm-polar an d B E '( r ~ ) is of finite type in #(£?', £7) for each r > 0. 2) However, property ii’) of Theorem 2.5 does not imply in general that i?E<(r) is c'-compact in a ( E \ E ) for each r > 0. Indeed, observe that if E has property ii’) of Theorem 2.5, then Be<[t) is c'compact if and only if r G |A"| (recall th at for each x G E, there is ƒ G B e ^ v ) such that | / ( i ) | = sup||9||<r |</(x)| = r||a:||). 3) One could think of considering also the property is c'-compact in cr(E\ E ).” B ut this possibility is not interesting at all because, if E ^ {0}, B e >(1~) = U e ^ I ) 1 is never c'-com pact in a ( E \ E ) . 4) The last part of Rem ark 2.7.1 yields the following natural question. P r o b Jem . Let E be an almost Hilbertian space. Docs it imply that £/?/(r) is of finite type for each r > 0? (Observe th at if E is norm-polar the converse is true. Indeed, if B e *(r ) is finite type then B ^ ( r " ' ) = B e ‘( i'Y is finite type. Now, apply Remark 2.7.1). 3. S T A B I L I T Y P R O P E R T I E S A N D E X A M P L E S Following [9], if I is an index set and {Ü?»}»e/ is a family of Banach spaces over A*, by X i z i E we will denote the set. of all elements a of the cartesian product f l t ' e / ^ or which the set {II«; I : i G /} is bounded. This Xi ç j Ei is a Banach space endowed with 46 the norm ||a|| — s u p | | < i i | | > T he elements a of f lig / ^ i ^or which, for each e > 0, the set {i G I : Ha» H > z} is finite form a closed subspace of Xig/JE,, denoted by Then, we have T h e o r e m 3.1. i) A subspace of a norm-polar (resp. a Hilbertian, an almost Hilbertian) subspace is a norm-polar (resp. a Hilbertian, an almost Hilbertian) space. ii) I f { E i } ieJ is a family of norm-polar (resp. Hilbertianf almost Hilbertian) spacest then g/.£?,* is again a norm-polar (resp. a Hilbertian, an almost Hilbertian) space. iii) I f {Ei}içr is a family of norm-polar spaces, then Xi ç j Ei is again a norm-polar space. I f in addition I is finite and every Ei (t G / ) w a Hilbertian (resp. an almost Hilbertian) space, then Y lisi 13 &gain a Hilbertian (resp. an almost Hilbertian) space. iv) If E is a norm-polar (resp. a Hilbertian, an almost Hilbertian) space and D is a finite-dimensional subspace of E, then E / D is again a norm-polar (resp, a Hilbertianj an almost Hilbertian) space. P r o o f . We prove i), ii ) and iv) for norm-polar spaces. Similar proofs work for H ilbertian and almost H ilbertian spaces. i) Let E be a norm -polar space and let M be a subspace. For each x £ M ~ {0}., [xj is almost orthocom plem ented in E s and hence in M . By Theorem 1.2, M is norm -polar. ii) Let x = ( l i ) j g / G x 0 anc^ ^ * € (0,1) be given. T here is a j E l such that ||rryII = ||.t||. Also, since Ej is norm-polar, [xj] has a i-orthogonal complement Sj in Ej. Take S = where Si = Ei if i ^ j . Then, for each s = (-s*)*e / € x 4 .s|| = max ||x; + s;|| > H#) 4 and so, S' is a ^-orthogonal complement of [æ] in || > = *11^ Now, apply Theorem 1.2. iii) Let G = X i ç j E i and let x = (ffi)ie/ € G, x ^ 0 and let e > 0 be given. We have to show th at there exists an ƒ G Gf with ||ƒ || < 1 such th a t ||æ|| —e < |f ( x ) \ . For that, let j G I be such th at ||x|| — e/2 < \\xj\\. Since Ej is norm-polar, there is f j G E j with \\fj\\ < 1 such th a t \\xj\\ - e/2 < |/ j ( x ;-)|. Then ƒ : G —* A", y = (y ^ ie / satisfies the required conditions. f j ( Vj ) Now, assume th a t I is finite. Then, the conclusion follows directly from ii). iv) Let x G E / D , x / 0 and let t G (0,1) be given. There is y G E such th a t 7T(y) = x (where tt : E —►E / D is the canonical surjection). Since E is norm -polar, D 4 [y] ha 5 a ^-orthogonal complement H in E (Theorem 1.2). Then, for each h G H , jr(/i) - x|| = inf \ \ h - y - d\\ > t inf ||y - d\\ = <||ir(y)|| dÇD afcD and so tt( H) is a t -orthogonal complement of K x . Now, the norm -polarity of E / D again follows by Theorem 1.2. 47 Rem arks 3.2. 1) The product x , e/£,' of a family of Hilbertian (almost H ilbertian) spaces is n o t always a Hilbertian (almost Hilbertian) space. E x a m p l e : Clearly K is a Hilbertian (and hence almost H ilbertian) space. However, £°° = x nÇrs|A' is not almost Hilbertian (its ‘o pen ’ unit ball is not weakly closed, see [3 ]). The class of norm-polar (resp. Hilbertian, almost Hilbertian) spaces is not closed for forming of quotients. 2) Indeed, for every Banach space B one can make a quotient m ap cQ( I ) — ►E if ƒ has adequate cardinal. Now, the conclusion follows by [9], Lem m a 4.35(ii), It is well-known th at for norm-polar Banach spaces E and F y their tensor product E ® F is also a norm-polar Banach space ([9], Corollary 4.34). To study the stability of the Hilbertian and almost Hilbertian property under the forming of tensor products we need the following preliminary result. L e m m a 3 . 3 . Let E , F be Banach spaces over A”, let D , S be closed subspaces of E and let G , T be closed subspaccs of F . Suppose that D X S and G X T. Then D § F + E®G X SgT. ■ P r o o f . By Lemma 4.30.ii) of [9] we may assume th a t E , F are of countable type. Let x G D<8>F + E®G, y G S § T and t G (0,1). We shall prove th a t ||x - y|| > <||y||, E has a ^-orthogonal base {e* : i G A^} where A g C N , such th a t {Ei : i G Ad} is a base for D for some Ad C Ag, and such th a t {e,- ; i G As} is a base for S for some A s C A e , where A s D A p = 0 ([9], T heorem 3.16). Similarly F has a i-orthogonal base {ƒ,■ : i G A/?} where A F C N, such th a t {ƒ■ : * 6 Ac} is a base for G for some Ag C A f , and such th a t {/j : i G Ar} is a base for T for some A^ C Ap, where A7- Pi A q = 0 . Then, {e, ® ej : ( 1, 7 ) G A s X Ap} is a i-orthogonal base for E ® F . Also, we can expand the elements .t and y as follows. »6A11 je Ar *6A£ Ag «€A5 >€At G K for all i, j ). Since (A s x A t ) n ( ( A d x A e ) U (A # x A g )) = 0 , 48 we have ® Cj\\ > t||y||. llx ~ v \\> i max Ic Aj je At T h e o r e m 3 . 4 . Let E , F be non-zero Banach spaces over A \ Then, E ® F is a Hilbertian (resp. an almost Hilbertian space) if and only if E and F are Hilbertian (resp. almost Hilbertian) spaces. P roof» We prove the result for Hilbertian spaces. For the case of almost H ilbertian spaces the proof is similar. First, suppose th a t E , F are Hilbertian spaces. Let x 6 E ® F , x ^ 0. We can write OO X ~ Y ] e" ® fn where e„ G F and ƒ„ € F for all n 6 N ([9]t Lemma 4.30). T here is an m (E N such th a t m \\'r Set and T = (IV ) / « i i < imi* H=l Since 5 , T are Hilbertian spaces (Theorem 3.1) we have th at 5 , T have orthogonal bases (Proposition 3.5,iv)) and so S ® T has also an orthogonal base ([9], Exercise 4.R.i), Hence, E n a s] e» ® /»] orthocom plem ented in S® 7\ Dy (IV) it is enough to prove th at £ ® T is orthocom plem ented in E ® F . To see that, let D be an orthogonal complement of S in E and let Q be an orthogonal complement of T in F. Then, by the previous lemma D<g)F + E ® G is an orthogonal complement of S® T. Now, suppose that E ($ F is a Hilbertian space. Since E and F can be isometrically * fc* identified with subspaces of E ® F h we conclude that E and F are H ilbertian spaces (Theorem 3.1.i)). In [4] and [9| we can find several examples of spaces which are (and which are not) norm-polar. The next result gives us some examples of Hilbertian and alm ost Hilbertian spaces and their relation to norm-polar spaces. P r o p o s i t i o n 3.5. i) Every Hilbertian space is almost Hilbertian. ii) Every almost Hilbertiaii space is a norm-polar space, iii) Every Danach .space v)ith an orthogonal base is a Hilbertian space. 49 iv) I f E is a Banach space of countable type, then E is Hilbertian if and only if E has an orthogonal base. v) Every finite-dimensional space is an almost Hilbertian space, vi) I f {Di}iç.[ is a family of finite'dimensional spaces then w an almost Hilber tian space. vii) I f E is an infinite-dimensional Banach space for which there are no non~zero orthogonal elements, then E is not an almost Hilbertian space. viii) There are almost Hilbertian spaces which are not Hilbertian. ix) There are norm-polar spaces which are not almost Hilbertian. Proof. i), v) and vii) are obvious. ii) Let D be a one-dimensional subspace of E and let ƒ G D ' with |ƒ | < || • || on D. Since E is almost Hilbertian *there exists a closed subspace M of E of finite codimension such that D is ortho complemented in M . Also, since M is almost orthocomplemented ([1], proof of Theorem 4.7.i) =*>- ii)), we conclude th at for every £ > 0 there exists ƒ € E ! extending ƒ with \f\ < (1 + e)|| • ||. Hence, E is norm-polar. For iii) and iv) see [9], vi) It is a direct consequence of v) and Theorem 3.1. viii) If E is a Hilbertian space (e.g. cq) and is th e two-dimensional space appearing in Example 1.1, it follows by the above properties and by Theorem 3.1 th at E © A';* is an almost Hilbertian space which is not Hilbertian. ix) Let F =s (c0, JV) be the Banach space of countable type considered in Example 1.1. By Theorem 3.16 of [9] F is norm-polar. By vii), F is n o t almost Hilbertian. R e m a r k 3,6. From considering the properties iii) and iv) of Proposition 3.5, the following question arises in a natural way. P r o b l e m . Does every Hilbertian space have an orthogonal base? Now, we are going to apply the above results to give some examples of norm-polar (resp. Hilbertian, almost Hilbertian) spaces consisting of some spaces of vector-valued continuous functions. For a HausdorfF zerodimensional topological space J ^ 0 a n d a Banach space E we define Cfc(-Y,J£): The space of all bounded continuous functions X E t endowed with the supremum norm, P C ( X ) E ) (resp. P ( X }E)): The space of all continuous functions ƒ : X —* E for which / ( X ) is precompact (resp. compactoid), endowed with the supremum norm. 50 W hen E = K we will write C b ( X ) , P C ( X ) and P ( X ) instead of C ^ X . K ) , P C ( X , K ) and P { X j q . Observe th at C b{ X) = P ( X) . It is straightfoi'ward to verify that Ch [ X, E) (resp. P C ( X t E)^ P ( X ^ E ) } is a normpolar space if and only if E is polar. Also, as in exercise 4.R of [9] and Theorem 1.3 of [2] one can easily prove th a t P C ( X ) % E (resp. P ( X ) ® E ) is isometrically isomorphic to P C ( X }E) (resp. P ( X , E ) ) . On the other h an d , since P C ( X ) has an orthogonal base ([9], Corollary 5.23) we have that P C ( X ) is a Hilbertian (and hence almost Hilbertian) space (Proposition 3.5.iii)). So, as a direct consequence of Theorem 3.4 we conclude: P r o p o s itio n 3.7, The following are equivalent. i) P C ( X , E ) is a Hilbertian (resp. an almost Hilbertian) space. ii) E is a Hilbertian (resp. an almost Hilbertian) space. The picture changes when we consider Cft(Ar, E ) and P ( X , E ) : P r o p o s i t i o n 3 . 8 . The following are equivalent, i) Cb ( X, E) is a Hilbertian (resp. an almost Hilbertian) space. ii) P ( X , E ) is a Hilbertian (resp. an almost Hilbertian) space. Hi) X is pseudocow.pact and E is a Hilbertian (resp, an almost Hilbertian) space. Proof, i) => ii) It follows from Theorem 3,l.i). ii) => iii) If-V is not pseudocompact we can find a countable infinite clopen partition X = U n -Ym. Choose e £ E — (0} and define T : £°° —►P ( X } E ) by tiie formula T(<yi , « 2 , . • •)(£) = OLne if n e N, x Ç X „ , We sec that ||Trv|| = ||tv||o&||e|| f°1' fV = (fti » • - •) £ ^°° so ifi a H ilbertian (resp. almost Hilbertian) space, which is a contradiction (see [3]). iii) => i) One verifies that, if X is pseudocompact, then Cf , (XyE ) = P C { X , E). Now apply Proposition 3.7. REFERENCES [lj Perez-Garcia, C.: Semi-Frcdhoim operators and the Calkin algebra in p-adic analysis M I, Bull. Soc. Math. Belg. Vol. XL1I (ser. B), 69-101 (1990). [2] Perez-Garcia, C., Schikhof, W.H.: Tensor product and p-adic vector valued contin uous functions, preprint. 51 [3] Perez-G ar ci a, C., Sch ik hof, W. H.; p-Adic ortho complemented subspaces in £°°. R eport 9313, M athematisch In s titu u t, Katholieke Universiteit, Nijmegen. Netherlands (1993). The [4] Schikhof, W.H.: Locally convex spaces over non-spherically complete valued fields I II, Bull. Soc. Math. Belg. Vol. X X X V III (ser. B), 187-224 (1986). [5] Schikhof, W.H.: Ultrametric com pactoids of finite type, Report 8634, Mathema tisch Instituut, Katholieke Universiteit, Nijmegen, T he Netherlands (1986), 1-19, [6] Schikhof, W.H.: Weak c'-compactness in p-adic Banach spaces, Report 8648, M athem atisch Instituut, Katholieke Universiteit, Nijmegen, The Netherlands (1986), 1-13. [7] Schikhof, W.H.: A connection between p-adic Banach spaces and locally con vex compactoids, Report 8736, M athem atisch Instituut, Katholieke Universiteit, Nijmegen, T h e Netherlands (1987), 1-16. [8] Schikhof, W .H., van Rooij, A.C.M.: O pen problems. In: p-adic functional aualysis, edited by J.M. Bayod, N. De Grande - De Kimpe and J. Martiuez-Maurica, Marcel Dekker, New York, 209-219 (1991). [9] Van Rooij, A.C.M.: Non-archimedean Functional Analysis, Marcel Dekkcr, New York (1978). [10] Van Tiel, J.: Esembles pseudo-polares dans les espaces localement A'-convexes, Indag. Math. 28, 369-373 (1966). 52

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