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If is a topological vector space and if this convex absorbing subset is also a bounded subset of , then the absorbing disk := | | = will also be bounded, in which case will be a norm and (,) will form what is known as an auxiliary normed space. If this normed space is a Banach space then is called a Banach disk.
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, the quotient of subspace theorem is an important property of finite-dimensional normed spaces, discovered by Vitali Milman. [1] Let (X, ||·||) be an N-dimensional normed space. There exist subspaces Z ⊂ Y ⊂ X such that the following holds:
Indeed, the elements of define a pointwise bounded family of continuous linear forms on the Banach space := ′, which is the continuous dual space of . By the uniform boundedness principle, the norms of elements of S , {\displaystyle S,} as functionals on X , {\displaystyle X,} that is, norms in the second dual Y ″ , {\displaystyle Y'',} are ...
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.
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 ‖ + ‖ + ‖ ‖ + ‖ ‖.
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 ...
The collection of all bounded sets on a topological vector space is called the von Neumann bornology or the (canonical) bornology of .. A base or fundamental system of bounded sets of is a set of bounded subsets of such that every bounded subset of is a subset of some . [1] The set of all bounded subsets of trivially forms a fundamental system of bounded sets of .