Saturating constructions for normed spaces

Saturating Constructions for Normed Spaces

We prove several results of the following type: given finite dimensional normed space V there exists another space X with log dimX = O(log dimV ) and such that every subspace (or quotient) of X, whose dimension is not “too small,” contains a further subspace isometric to V . This sheds new light on the structure of such large subspaces or quotients (resp., large sections or projections of conve...

Saturating Constructions for Normed Spaces II

We prove several results of the following type: given finite dimensional normed space V possessing certain geometric property there exists another space X having the same property and such that (1) log dimX = O(log dimV ) and (2) every subspace of X, whose dimension is not “too small,” contains a further wellcomplemented subspace nearly isometric to V . This sheds new light on the structure of ...

NORMED HYPERVECTOR SPACES

The main purpose of this paper is to study normed hypervector spaces. We generalize some definitions such as basis, convexity, operator norm, closed set, Cauchy sequences, and continuity in such spaces and prove some theorems about them.

-approximation in normed spaces

Normed Vector Spaces

A normed vector space is a real or complex vector space in which a norm has been defined. Formally, one says that a normed vector space is a pair (V, ∥ · ∥) where V is a vector space over K and ∥ · ∥ is a norm in V , but then one usually uses the usual abuse of language and refers to V as being the normed space. Sometimes (frequently?) one has to consider more than one norm at the same time; th...