Normed proper quasilinear spaces

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...

ON n-NORMED SPACES

Given ann-normed space withn≥ 2, we offer a simple way to derive an (n−1)norm from the n-norm and realize that any n-normed space is an (n−1)-normed space. We also show that, in certain cases, the (n−1)-norm can be derived from the n-norm in such a way that the convergence and completeness in the n-norm is equivalent to those in the derived (n− 1)-norm. Using this fact, we prove a fixed point t...

Cone normed spaces

In this paper, we introduce the cone normed spaces and cone bounded linear mappings. Among other things, we prove the Baire category theorem and the Banach--Steinhaus theorem in cone normed spaces.