On Y -closed subspaces of X, for Banach spaces X ⊂ Y ; existence of alternating elements in subspaces of C(J)

Jürgen Voigt

Vol. 81 (1979), No. 1, 253–266

DOI: 10.2140/pjm.1979.81.253

Abstract

If X ⊂ Y are Banach spaces, with continuous embedding, we consider property (P3): If L ⊂ X is a closed subspace of Y , then L is finite dimensional. If the embedding X↪Y is compact (property (P1)), then (P3) follows. It is shown that (P1) implies also (P2): In (P3) the dimension of L can be estimated from above in terms of the norm of the mapping id : (L,∥⋅∥_Y ) → (L,∥⋅∥_X). For some examples which are known to satisfy (P3) but not (P1), we show that also (P2) is valid.

The main tool for the proof of (P1) ⇒ (P2) is the existence of “alternating” elements in subspaces of R^k and C[0,1]. In order to obtain such elements we investigate the structure of certain subsets of the unit cube in R^k.

Mathematical Subject Classification 2000

Primary: 47B05, 47B05

Secondary: 46A50

Milestones

Received: 21 March 1978

Published: 1 March 1979