Sentence examples for that normed from inspiring English sources

In this paper we show that normed structures which can be axiomatized in positive bounded logic (in the sense of Henson and Iovino) admit proof-theoretic metatheorems (as developed by the second author since 2005) on the extractability of explicit uniform bounds from proofs in the respective theories.

It is clear that normed linear spaces, l p (or L p ) spaces ( p > 0 ), l ∞ (or L ∞ ) spaces, Hilbert spaces, Banach spaces, hyperbolic spaces, ℝ-trees and CAT ( 0 ) spaces are examples of b-metric spaces.

Schauder fixed point theorem states that any continuous mapping of a nonempty convex subset of a normed space into a compact set of that normed space has a fixed point [36, Theorem  4.1.1].

We show that every normed spaceEwith a weakly locally uniformly rotund norm has an equivalent locally uniformly rotund norm.

It is clear that all normed linear spaces are hyperbolic in this sense.

One can check that the normed space ((F_{(nu,beta,mu,k)}^{d},|cdot|_{(nu,beta,mu,k)})) is a Banach space.

Remark 1 In the case that the normed vector space ( E, ∥ ⋅ ∥ ) is complete, if (2) holds then the mapping B is well defined.

In 1970, Takahashi [1] introduced the notion of convexity in metric spaces and proved that all normed spaces and their convex subsets are convex metric spaces.

It was shown in [1] that a normed space is an inner product space if and only if it is a Ptolemy space.

In this paper, we use a locally convex space as a target set for a cone pseudo-metric, which is more general that a normed space.

It is well known that a normed space is uniformly convex (smooth) if and only if its dual space is uniformly smooth (convex).