An extension of the normed dual functors

Abstract

By means of the direct limit technique, with every normed space X it is associated a bidualic (Banach) space $\tilde{X}(D^2(\tilde{X})\cong \tilde{X}$ - called the hyperdual of $X$) that contains (isometrically embedded) $X$ as well as all the even (normed) duals $D^{2n}(X)$, which make an increasing sequence of the category retracts. The algebraic dimension dim $\tilde{X}$ = dim $X$ (dim $\tilde{X}$ = $2^{{\aleph }_0}$ ), whenever dim $X\ne {\aleph }_0$, (dim $X={\aleph }_0$). Furthermore, the correspondence $X\mapsto \tilde{X}$ extends to a faithful covariant functor (called the hyperdual functor) on the category of normed spaces.

References in corpus (1)

• The quotient shapes of normed spaces and application