By Ivan Singer
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Extra resources for The theory of best approximation and functional analysis
Example text
THE THEORY OF BEST APPROXIMATION AND FUNCTIONAL ANALYSIS 29 (f) Let us consider now closed linear subspaces G of finite codimension. 1 one obtains the following result, due to A. L. Garkavi (see [168, p. 296]). 7. A closed linear subspace G of codimension n of a normed linear space E, say G = (xeEI/^x) = • • • = fn(x) = 0} (with f l , - - - ,fneE* linearly independent), is a semi-Chebyshev subspace if and only if for every /e G1\{0} either the set Jt f of all maximal elements off is empty or Jf f is of dimension r = r(f) ^ n — I and contains r + 1 elements x 0 ,x 1 ?
Morris and J. E. Olson [8]); on the other hand, it is clear that the unit vector {1,0,0, • • • } in the complex spaces /1 or l^ spans a one-dimensional interpolating subspace. 6. Almost Chebyshev subspaces. &-semi-Chebyshev and A>Chebyshev subspaces. Pseudo-Chebyshev subspaces. We shall consider now some generalizations of semiChebyshev and Chebyshev subspaces. 5. A set G in a metric space E is called an almost Chebyshev set if the set of all x e E for which ^G(x) does not consist of a single element forms a set at most of the first category in E.
NG is closed. The proofs are straightforward (see [168, pp. 140-142 and 390]). Part (i) (and (vii)) shows that in the particular case when G is a Chebyshev subspace, the metric projection nG is indeed a (nonlinear closed) projection of E onto G. 2. Continuity of metric projections. In the present section we shall be concerned with Chebyshev subspaces G for which the metric projection nG is continuous. We know of no convenient short term for such subspaces in the literature (some THE THEORY OF BEST APPROXIMATION AND FUNCTIONAL ANALYSIS 41 authors use the term EU*-subspace) and we shall not introduce any special term for them.