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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 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 ...
A T 4 space is a T 1 space X that is normal; this is equivalent to X being normal and Hausdorff. A completely normal space, or hereditarily normal space, is a topological space X such that every subspace of X is a normal space. It turns out that X is completely normal if and only if every two separated sets can be separated by neighbourhoods.
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space.Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space.
A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a seminormed vector space. The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm". [1]
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.
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 ...
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 ...