*PM : necessary and sufficient conditions for a normed vector space to be a Banach space, id = 9751 new !-- WP guess : necessary and sufficient conditions for a normed vector space to be a Banach space-- Status:
42.
*PM : necessary and sufficient conditions for a normed vector space to be a Banach space, id = 9751 new !-- WP guess : necessary and sufficient conditions for a normed vector space to be a Banach space-- Status:
43.
One can also use this lemma to demonstrate whether or not the normed vector space X is finite dimensional or otherwise : if the closed unit ball is compact, then X is finite dimensional ( we proceed by contradiction to this proof this results ).
44.
Every bounded linear transformation \ mathsf { T } from a normed vector space X to a complete, normed vector space Y can be uniquely extended to a bounded linear transformation \ tilde { \ mathsf { T } } from the completion of X to Y.
45.
Every bounded linear transformation \ mathsf { T } from a normed vector space X to a complete, normed vector space Y can be uniquely extended to a bounded linear transformation \ tilde { \ mathsf { T } } from the completion of X to Y.
46.
A continuous embedding of two normed vector spaces, X \ hookrightarrow Y is called " cocompact " relative to a group of isometries G on X if every G-weakly convergent sequence ( x _ k ) \ subset X is convergent in Y.
47.
These include theorems about compactness of certain spaces such as the Banach Alaoglu theorem on the weak-* compactness of the unit ball of the dual space of a normed vector space, and the Arzel? Ascoli theorem characterizing the sequences of functions in which every subsequence has a cellular automata.
48.
If the earlier completion procedure is applied to a normed vector space, the result is a Banach space containing the original space as a dense subspace, and if it is applied to an inner product space, the result is a Hilbert space containing the original space as a dense subspace.
49.
*Conversely, if V is a normed vector space with the norm \ | \ cdot \ | then there always exists ( maynot be unique ) a semi-inner-product on V that is "'consistent "'with the norm on V in the sense that
50.
In simplest terms, the Kantorovich inequality translates the basic idea of the triangle inequality into the terms and notational conventions of linear programming . ( See vector space, inner product, and normed vector space for other examples of how the basic ideas inherent in the triangle inequality line segment and distance can be generalized into a broader context .)
