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In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets.
In mathematics, a strictly convex space is a normed vector space (X, || ||) for which the closed unit ball is a strictly convex set. Put another way, a strictly convex space is one for which, given any two distinct points x and y on the unit sphere ∂B (i.e. the boundary of the unit ball B of X), the segment joining x and y meets ∂B only at ...
However, every finite dimensional normed space is a reflexive Banach space, so Riesz’s lemma does holds for = when the normed space is finite-dimensional, as will now be shown. When the dimension of X {\displaystyle X} is finite then the closed unit ball B ⊆ X {\displaystyle B\subseteq X} is compact.
In mathematics, a uniformly smooth space is a normed vector space satisfying the property that for every > there exists > such that if , with ‖ ‖ = and ‖ ‖ then ‖ + ‖ + ‖ ‖ + ‖ ‖.
Every normed vector space can be "uniquely extended" to a Banach space, which makes normed spaces intimately related to Banach spaces. Every Banach space is a normed space but converse is not true. For example, the set of the finite sequences of real numbers can be normed with the Euclidean norm , but it is not complete for this norm.
All Banach spaces and Fréchet spaces are F-spaces. In particular, a Banach space is an F-space with an additional requirement that (,) = | | (,). [1]. The L p spaces can be made into F-spaces for all and for they can be made into locally convex and thus Fréchet spaces and even Banach spaces.
In mathematics, Schur's property, named after Issai Schur, is the property of normed spaces that is satisfied precisely if weak convergence of sequences entails convergence in norm. Motivation [ edit ]
The space of continuous real valued functions with compact support on a topological space with the pointwise partial order defined by when () for all , is a Riesz space. It is Archimedean, but usually does not have the principal projection property unless X {\displaystyle X} satisfies further conditions (for example, being extremally ...