Question

Quotient space and decompsition of banach space as a direct sum

Answer 1

It turns out that a decomposition as in (1) exists for a given separable Banach space if and only if it is possible to decompose the space into a sum of two infinite-dimensional and infinite-codimensional subspaces (Theorem 2.1). Let us note that not every separable Banach space has a decomposition (1).

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