On /с-interpolating subspaces

ROCZNIK.I POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXIV (1984)

W.

(Poznan)

On /с-interpolating subspaces

1. Introduction. For the space C [a, b] there is a strict correspondence between the notion of Нааг subspace and the unique solution problem in the best approximation from a finite dimensional linear subspace. In [1] D. Alut, F. Deutsch, P. Morris and J. Olson carried over this notion to normed real spaces under the name of interpolating subspace. Earlier, this notion appears implicite in I. Singer’s paper [3] in connection with a characterization of the dimension of the set PM{x) consisting of all nearest to x elements among the elements of a subspace M:

T

([4], p. 239). Let M = [ x 1? ... , x„] be an n-dimensional linear subspace of normed linear space X, spanned on x 1? ..., x n. Let 0 ^ к < n. In order that M be a k-Chebyshev subspace, i.e. sup (dim PM(x): x e l } ^ к, it is sufficient that :

(Hk) for any n — k linearly independent extremal points (pt , ..., (p„-k of the unit ball U{X*)

( 1 . 1 ) rank (<Pj(Xifyz\—'tH n_k = n~k ,

As it will be seen later, finite dimensional subspaces for which condition (Hk) holds are exactly /(-interpolating subspaces.

It is clear that describing such subspaces and spaces that admit them is of great interest. In this direction much was done in [1], for the case к = 0, i e. for interpolating subspaces.

Here we shall present a few basic results concerning the concept of k- mterpolating subspace. Namely, we study characterization theorems, some operational rules, a connection with the so-called Ф-interpolative subspaces (see e.g. [ 2 ]) and finally a problem of non-existence of /(-interpolating subspaces in spaces whose duals are (k + l)-strictly convex. For к = 0 this is concerned with a weaker version of a similar result from [1]. An example of /(-interpolating subspace which is not '/-interpolating for some / < к is also given.

Let us agree on some notations. If not otherwise stated, M denotes a

finite dimensional proper (linear) subspace of a real normed space X. A* is

the dual space to X. ext A denotes the set of all extreme points of a set

A, t/(2Q-closed unit ball in X, M 1 (M1) the anihilator subspace in X*(X) for M с X (M cz X *), [A] denotes the linear hull of a set A.

2. Characterization of /.-interpolating subspaces. We begin with the following definition that is basic for our purposes.

D

2.1. An n-dimensional (proper) subspace

с

is called k- interpolating subspace, where 0 ^ к < n, if for each set of n — k of linearly independent elements q>t , ..., (p„-k eext U

and each set of n — k real scalars a l5 ..., an_k there is an element

such that

— oq, i = 1, ...

..., n — k.

The О-interpolating subspace is an interpolating subspace in the sense of R em ark . The definition can be put into another equivalent form: an n- dimensional subspace M <z X is called /^-interpolating, where 0 < к < n, if for every subset a <= ext U(X*) of algebraic dimension dim <7 ^ n — k and for every x e X there is y e M such that

Moreover, in this definition one may consider subsets a c ext U (X *) of dimension n — k only.

The equivalence mentioned is justified by the following simple obser­

vation: given a = \(pi , . . . , (p„-k} being linearly independent and ofj, ..., a„_ke/?, then there always is x e X for which (Pi(x) — ah i = 1, ...

..., n — k.

Let us note that since M is a finite dimensional, say n-dimensional, proper subspace of X, ext U (X*) contains at least n + 1 linearly independent extreme points.

P

2.1. Let 0 ^ к

n, where n is the dimension of a given subspace M c X . The following statements on M are equivalent.

(a) M is a к-interpolating subspace.

(b) Given any basis x {, ..., x n of M, for every linearly indeperulent elements </>l5 ..., <p„_keext U (X*) rank M„_k>„ = n - k , where

(c) (The generalized Haar condition.) Every set of к 4-1 linearly indepen­

dent elements y t , ..., yk+i e M has in ext U(X*) at most n — k — 1 common linearly independent “zeroes”, i.e. such elements (pt , ..., <p,eext U (X*) being linearly independent, where l ^ n — k —1, for which q>j (xi) = 0 j =

(d) Each n — k elements o f ext U (X*) are linearly independent exactly when their restrictions to M are linearly independent.

[ 1 ].

(

2

1

<p(x) = <p(y) for all (pea.

(e) Ta [M ] = Rn k for every linearly independent subset a

= {(p1, ..., (pn- k} c ext U (X*), where Тв: X -> Rn~k is a linear map of the form 7^(x) —(<pl (x), . . (pn- k(x)). When this is the case, Ker T„

— [<Pl» • • •» <Pn-*]±-

The proof is an immediate consequence of the definitions and of some well-known algebraic dependences, so it is omitted.

Condition (b) is nothing but condition (Hk) in Section 1. Moreover, the equivalence of (b) and (c) appears also, together with a few others, in the context of the theorem from the first section, in monograph [4], p. 239.

In the next theorem we shall give another collection of properties equivalent to the first one in the above proposition.

Th e o r e m

2.1. Let 0

к

n, where n is the dimension of a given subspace M c= X . The following statements on M are equivalent.

(a) M is a к-interpolating subspace.

(f) For every subset о c: ext U (X*), with dim a — n — k, M L n M = Ï 0 }.

(g)

1 n ( у M ) = JO}, where the sum is taken over all (n — k)-dimensional

a

subsets a c: ext U (X*).

(h) For every о

ext U (X*) of dim o = n — k the inequality holds:

dim (M n ^ k.

R em ark. The theorem holds true even for a cz ext U(X*) being of dimension at most n — k only, or for a = [<p l5 ..., (p„-k) <= ext U(X*) with dim о = n — k. Let us also mention that the equivalence of (a), (f), (g), for к

= 0 , are given in [ 1 ], where о is of the form {<plt ..., (p„}.

P roof. The equivalence of (f) and (g) is clear because of M x n ( U M ) a - y (M x n [<

]) = [0] . To prove that (a) is equivalent to (f) we show the

<T

converse equivalence. To do it let <r <= ext U (2f*) be such that (f) does not hold, and <plt ..., <p„_k be any linearly independent subset in a. A functional

n — 1

<pe[«r] is of the form (p = £ a. t/),- and it is in M 1 exactly when M x- k<na = 0 i = 1

with non-zero a = (a1? ..., a„-*)1. But the latter means that rank M„_k>n <

п — к — 1 for some linearly independent cp1, ..., <p„_keext U (X*) and for any basis xb ..., x„ of M. Hence, condition (b) from Proposition 2.1 does not hold, so it is true also for condition (a). Next, to prove the equivalence of conditions (a) and (h) use the same scheme, i.e. prove the converse equival­

ence of conditions (b) and (h). For such a purpose it suffices to list the following statements equivalent to each other: dim (Af n [<р 19 ..., <p„_k]) >

k + 1 (for given linearly independent <pl , ..., (pn- ke o ); a system of equations Mn-k,na = 0, a e R n has at least k + 1 linearly independent solutions; rank Mn~k,„ < n - k - 1 .

A corollary below collects further group of conditions equivalent to

condition (a). They are simply a dual version to those of the theorem.

Co r o l l a r y

2.1 Let 0 ^ к < n, where n is the dimension of a given subspace M

X . The following statements on M are equivalent.

(a) M is a к-interpolating subspace.

(i) For о a ext U (A*), with dim о ^ n — k, X = M + [a]L.

(j) For every a cz ext U(X*), with dim о ^ n — k, there is a subspace

N

cr M such that X = A 0[er] dim JV = dim a.

(k) X — (M 1 n ((J [ff]))±, о

ext U(X*), dim о ^ n — k.

G

R em ark . As in the above theorem the corollary holds true even for <

of the form {<px, ..., (p„~k}, or for a to be any subset of ext U(X*) — in both cases of dimension n — k.

P ro o f. The proof is based on a number of well-known duality formu­

lations. Equivalence of (g) and (k) is evident, (f) is equivalent to (i), because X = {0}± = (M 1n (VJ)! = œ ( M ± u [cr]x) = M -f [cr]x; here we make use of finite dimensions of M and o. Condition (j) implies condition (i). To prove the converse implication we first note that M = Af©(M + [ff]x) for some subspace

M. Hence X — N 0[<r]i . Now we prove that dim

= dim o.

Choose any base of

say xl 9 . . . , x m. Of course, m ^ n. The above decomposition of X means that for every x e X there is exactly one element

m m m

u =

such that x — £ а, хг e [er]j_, i.e.

£ ai xi) = 0

i = 1 i = 1 i = 1

for any linearly independent system q>x, ..., (p ^ a with / = dim a. Hence the system M t ma = b, where M, m is of the form (2.2) and b = (q>x (x), ..., <p,(x))T, has exactly one solution a e R m for every x e X . Therefore / = m, i.e. dim

= dim a.

Co r o l l a r y

2.2. For к, n, M as in the above theorem the following statements on M are equivalent.

(a) M is a к-interpolating subspace.

(b) For every

ext

dim

n — k, there is a subspace

M such that X* = IVх©[<r], and dim

= dim

Here, as in the above, о may be taken also (n — k)-dimensional only-even in the form {(px, ..., <pn- k}.

P ro o f. It suffice to prove the equivalence of (k) and (1). But this is an immediate consequence of the relations:

(ЛГ 1 + M )x = TOfJV 1 и И ) = (N n W J 1 = {O }1 = X*

W 1 n И = N x n O ] 1 = (N u H J 1

= (со (N u M J) 1 = (JV + M i)1 = X1 = {0}.

Here it was extensively used that M, [<r] arç finite-dimensional subspaces.

3. A connection with Ф-interpolative subspaces. Following [2] we have the definition of another type of linear subspaces of X, that may be infinite dimensional.

D

3.1. Let Ф be a linear subspace of X. A linear subspace L с X is said to be Ф-interpolative if for every x e X there exists one and only one y e M such that

( 3 . 1 ) (p(x) = q>(y) for all (реФ.

A connection of к-interpolating subspaces with Ф-interpolative subspaces is given in the following theorem.

T

3.1. A n-dimensional subspace M c= X is к-interpolating if and only if for every set a c ext U (X*), with dim о ^ n — к, M contains a [<r]- interpolative subspace N cz M such that dim N = dim a .

The theorem follows immediately from Corollary 2.1, by applying the equivalence of (a) and (j), and from the observation given in [2]: L с X is Ф- interpolative subspace if and only if X = L®Ф1 (-algebraic direct sum).

There are reasons for the study of a somewhat wider concept than Ф- interpolative subspace, namely Ф-interpolative subspace of rank k, where 0 < к < dim Ц for which it is assumed (by definition) that 0 ^ dim J x ^ к for each x e X . Here J denotes a map J: X - +2 L defined by (3.1), i.e.:

(3.2) J x = { y eL: х - у е Ф ±}.

P

3.1. Let Lbe a linear subspace of X and let Ф be a linear subspace of X*. Let 0 ^ к < dim L. Then L is Ф-interpolative of rank к if and only if X = L + Ф± and 0 ^ dim L n Ф L < k.

P roof. First, we note that J x Ф 0 for every x e X , is equivalent to the decomposition X = L + Ф^. Now, it is sufficient to show that dim J x

= dim Lr^Ф1 for each x e X , whenever J x Ф 0 . It is easy to see that x — J x

= (L + x ) n Ф± = Тп(Ф± —x) + x, hence dim J x = dim L n ^ ± — x). We shall prove that

(3.3) dim L n ^ - x ) ^ dim Т п Ф ±; x e J .

Assume for a moment the converse (strict) inequality in (3.3) for some x e X . Let ul , . . . , u s be a maximal afine independent subset in L n ^ j ^ — x), Of

5

course щ = Vj — x, i — 0, ..., s for suitable vi eФ1. Therefore, if £ a. i;, = 0

i = 0

s s s

and £ a, = 0, then a,(i;( —x) = 0 and £ a,- = 0. Hence af = 0,

« = 0 i = 0 i = 0

1 = 0, ..., s. Thus affine independence of щ implies the same for t?f. Further ui~ u 0 = vt — v0e L n Ф±, i = l , . . . , s . Hence, d i m L r ^ ^ s , and s = dim 1 п (Ф х- х ) that gives the contradiction.

Finally, to prove that in (3.3) a strict inequality is impossible let us note

that from 3ХФ 0 , for every x = u + v, u e L , геФ ± we have L n (Ф±-х)

— (L— u) n (Ф±- и ) =э Ь п Ф ± — и. Therefore dim L n (Ф i - x ) ^ dim L о Ф± and the proof follows.

Here we give only the following application of the above proposition and of equivalence (a) and (i) from Corollary 2.1.

Th e o r e m

3.2. An n-dimensional subspace M а X is к-interpolating with О ^ к < n if and only if it is [pointerpolative of rank к for each a c ext U (X *) with dim о ^ n —к .

4. Operations on к-interpolating subspaces. It is clear that if M cz X is a /(-interpolating m-dimensional subspace, then it is also an /-interpolating subspace for each l satisfying к ^ / < m. The converse is not true in general as shows an example in the last section. Hence, О-interpolating subspace is a /(-interpolating subspace for all 0 < к < m. Also, if M c N are m and n- dimensional subspaces, respectively, and M is /(-interpolating subspace, then N is an (n — m + /^-interpolating subspace. This follows from the definition or from the equivalence of (a) and (i) in Corollary 2.1.

Th e o r e m

4.1. Let M and N be linear subspaces of X of dimension m and n, respectively. Let dim M n N = r. I f M is a к-interpolating and N is an /- interpolating subspace, then M + N is an s-interpolating with

(4.1) s = min {n + k, m + l} — r.

P ro o f. For the subspaces M,

the formula holds (M +

=

According to the equivalence of (a) and (i) in Corollary 2.1 it suffices to prove that (M +

n [VJ = (Oj for some о c ext

with dim a ^ m + n — r — s.

Fix the case s = n + k — r. The above formula yields M 1 r \ N 1 n О ] = Ю!, because M is /(-interpolating and m — k — m + n — r — (n + k — r).

Co r o l l a r y

4.1. Suppose M, is kr interpolating subspace for i = 1, ..., m.

m

Then + Mf is s-interpolating with

i = l

m

(4.2) s = dim ( + M,)— max (dim Mj — kj).

i - 1 j =

m

P ro o f. The case m = 2 is that of Theorem 4.1. Let + Mf be s- i— 1

interpolating subspace with s given by 4.2. Applying the theorem for M m

= ( + M,) + M m+1 we obtain that it is s'-interpolating with

i - 1

m m

s' = min (dim M m+l +s, dim ( + M,) + /(m+i} —dim (Mm+ 1 n ( 4 M,)),

i = l i= 1

so that the rest of the proof is clear.

The next corollary is on improving the rank of an interpolating subspace.

Co r o l l a r y

4.2. Let M cz N a X . I f M is a к-interpolating subspace and N is an l-interpolating subspace, then N is an s-interpolating subspace with (4.3) s = min {/, dim N — dim M + k}.

Let us mention that for X = C(Q) a к -interpolating subspace is nothing else as a generalized version of Haar subspace. Moreover, it is known, cf. [4], p. 241, that in order that an «-dimensional linear subspace M cr C(Q) be a k-Chebyshev it is necessary and sufficient that condition (c) from Proposition 2.1 holds. Therefore, form the above corollary we obtain:

Pr o p o s it io n

4.1. Suppose

is a kr Chebyshev subspace of finite dimen- m

sion, where i = 1, ..., m. Then + is an s-Chebyshev subspace in C(fl) I = l

with s given by formula (4.2).

5. Existence of к-interpolating subspaces. In fact we shall give a theorem which is of negative character. This theorem is analogous to the result from [1]; Theorem 3.1.

We shall need the following lemma.

Le m m a

5.1. Let E={ ( p e U( X* ) : t p e M f (p(x0) = 1}, where a given point x 0e M has norm one, and M is any linear subspace of X. Then E is non­

empty, a)*-compact, convex subset of X * which is U (X*)-extremal.

P roof. By Hahn-Banach theorem the set E ф 0 . It is also convex and from the Alaoglou-Bourbaki theorem is also cu*-compact. Hence ext E Ф 0 . To show that E is £/(X*)-extremal subset — i.e. the relations q>i , (p2£U(X*), X(pl +(l — X)(p2eE, with 0 < A < 1 , always imply (pl ,(p2e E — fix two elements tp1, q>2e E and assume Xtp1 +(1 — À) cp2 eE . Since |jr/>,|! = 1, i = l , 2 , then \<Pi(x)\ ^ 1 for ||x|| = 1. Therefore, if <p,(x0) Ф 1 for some i, then in fact \(Pi(x0)\ < 1 for some i = 1,2. Hence we obtain l=|A<p 1 (.x0) + + (1 —X)(p2(x0)\ < A|<p 1 (xo)| + (l-A)|<p 2 (x0)l < 1 , i.e. the required contradic­

tion. Thus tpi(x0) = 1, i = 1, 2. Next, let us mention that ltpl +(1 —X)q>2 — 0 on M implies that (pt , (p2 take a value zero simultaneously. Now, assume for a moment that 0 < (pfm), i = 1, 2, for some m e M . Hence also 0 < <p,(m)4-E for suitable e < 0, where / = 1,2. From this we have 0 < (Xtpi +(1 — X)(p2)(m) + + £ = s < 0 which is impossible. Therefore (Pi(m) — 0, i — 1, 2, and the lemma is proved.

Th e o r e m

5.1. Let X* be {n~k + \)~strictly convex, dual to X. Then X does not contain any n-dimensional к-interpolating subspace, where к < n

< dim X.

R em ark. For a fixed к ^ 0 if X does not contain any «-dimensional k- interpolating subspace, where к < n < dim X , then X does not contain any l- mterpolating «-dimensional subspace with 0 ^ < k.

Prace Materaatyczne 24.1

P ro o f. Assume conversely that there is an n-dimensional subspace M

X, к

n

dim X, which is /с-interpolating. Consider a subset

E = {(peU(X*): (peM 1, <p(x0) = 1} of FXQ = {<peU(X*), <p(x0) = 1}.

From the lemma above and the well-known properties of extremal subsets it easily follows that ext E = E n ext U (X*) and that ext E Ф 0 . Further, by I.

Singer’s result, cf. [4], p. 128, X is (n — k + \) *- strictly convex exactly when the algebraic dimension of Fx** = \<peU(X*): X**((p) = 1} is at most n — k , for all x** eU(X**). Hence dim E ^ dim FX

^ n — k. Thus we have proved the existence of the set < 70 c: ext U (X*) with 0 ^ dim a0 < n — k and such that <r 0 c M x (and so [o-0] с M 1) — namely a0 = ext E. On the other hand, M is a к-interpolating subspace, so in view of the equivalence of (a) and (0 in Theorem 2.2, we have that for every a c ext U (X*) with dim <т < и — к, Мх п[(т] = {0}. Which contradicts the above information on <r 0 and the theorem is proved.

C

5.1. I f X* is strictly convex, dual to X, then X does not contain any к-interpolating subspacc of dimension n with k, n satisfying 0 ^ к

< n < dim X.

R em ark . A part of this corollary related to the case к — 0 is the result from [1], Theorem 3.1. Let us also mention that Theorem 5.1 does not agree for к = 0 with this result either, because our assumptions are of weaker form.

Finally, let us mention that if a normed space X contains some k- interpolating subspace than no extremal point of U (X*) is orthogonal to this subspace. This is a simple consequence of Theorem 2.1 — the equivalence of (a) and (f).

6 . Examples. Here we deal with the space X = C(T, R2) of all con­

tinuous functions from T = [0, 1] to R2 equiped with the Euclidean norm.

Consider the following linear subspace of C(T, R2)

This subspace is of dimension five. It is a well-known fact that for every

<pepxt U(C*(T, R2)), (p(f) = ?(f(t)) for some z e e x tU (R 2) and teT, where / eC(T, R2), so we may identify q> with a pair (

, t). Moreover r

Consider four extreme points of the form:

It is easy to show their linear independence. From equivalence of conditions (a) and (b), from Proposition 2.1, for n = 5 taking in turn к such that 5 — k — i, i = 4, 3, 2 we see that M is not a 1, 2, 3-interpolating subspace. Hence, it is not О-interpolating subspace, cf. the beginning of Section 4. On the other hand, taking any extremal point (p = t^ where + = 1, and teT, we deduce from the mentioned proposition that rank M 5_ 4 5 = 1, so M is 4-interpolating.

Quite similary one can prove that the linear subspace

N = t d - t )

0 V„)

which is of dimension four, is not ^.-interpolating for к — 0, 1, 2, 3. Thus we have obtained an example of the subspace N с M, where M is such as above, which is not /-interpolating while M is /с-interpolating, namely for к = 4.

R em ark. From the point of view of the theorem in Section 1 (cf.

remarks after Proposition 2.1) one can say that it is something simpler to work with к-interpolating subspaces when к is near to n — 1. But, on the other hand, it is better to know the smallest rank of k-Chebyshev subspaces.

References

[1] D. A lu t, F. D e u ts h , P. M o r r is, J. O ls o n , Interpolatitig subspaces in approximation theory, J. Approx. Th. 3 (1970).

[2] J. Mi lo t a, Interpolation in a Banach space, Czechosl. Math. J. 26 (1976).

[3] I. Si nger, On the set of best approximations of an element in a normed linear space, Rev.

Math. Pures Appl. 5 (1960).

H ] —, Best approximation in normed linear spaces by elements of linear subspaces, Publ. House Acad. Soc. Rep. Romania, Bucharest, and Springer-Verlag Berlin-Heidelberg-New York 1970.