Vol. 81, No. 1, 1979

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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

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 XY 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 Rk and C[0,1]. In order to obtain such elements we investigate the structure of certain subsets of the unit cube in Rk.

Mathematical Subject Classification 2000
Primary: 47B05, 47B05
Secondary: 46A50
Received: 21 March 1978
Published: 1 March 1979
Jürgen Voigt