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