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As the standard negator is used in the above definition of a t-norm/t-conorm pair, this can be generalized as follows: A De Morgan triplet is a triple (T,⊥,n) such that [1] T is a t-norm; ⊥ is a t-conorm according to the axiomatic definition of t-conorms as mentioned above; n is a strong negator
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.
The unit sphere can be replaced with the closed unit ball in the definition. Namely, a normed vector space is uniformly convex if and only if for every < there is some > so that, for any two vectors and in the closed unit ball (i.e. ‖ ‖ and ‖ ‖) with ‖ ‖, one has ‖ + ‖ (note that, given , the corresponding value of could be smaller than the one provided by the original weaker ...
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.
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.
The space of distributions, being defined as the continuous dual space of (), is then endowed with the (non-metrizable) strong dual topology induced by () and the canonical LF-topology (this topology is a generalization of the usual operator norm induced topology that is placed on the continuous dual spaces of normed spaces).
For example, () is the space of all sequences indexed by the integers, and when defining the -norm on such a space, one sums over all the integers. The space ℓ p ( n ) , {\displaystyle \ell ^{p}(n),} where n {\displaystyle n} is the set with n {\displaystyle n} elements, is R n {\displaystyle \mathbb {R} ^{n}} with its p {\displaystyle p ...
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 ...