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Banach space, normed vector spaces which are complete with respect to the metric induced by the norm; Banach–Mazur compactum – Concept in functional analysis; Finsler manifold, where the length of each tangent vector is determined by a norm; Inner product space, normed vector spaces where the norm is given by an inner product
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.
The Lebesgue space. The normed vector space ((,), ‖ ‖) is called space or the Lebesgue space of -th power integrable functions and it is a Banach space for every (meaning that it is a complete metric space, a result that is sometimes called the Riesz–Fischer theorem).
In contrast, infinite-dimensional normed vector spaces may or may not be complete; those that are complete are Banach spaces. The space C [a, b] of continuous real-valued functions on a closed and bounded interval is a Banach space, and so a complete metric space, with respect to the supremum norm.
Infinite-dimensional normed vector spaces, particularly spaces of functions, are studied in functional analysis. Completeness is particularly important in this context: a complete normed vector space is known as a Banach space. An unusual property of normed vector spaces is that linear transformations between them are continuous if and only if ...
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]
[note 6] [13] All norms on a finite-dimensional vector space are equivalent and every finite-dimensional normed space is a Banach space. [14] Every Banach space is a complete TVS. A normed space is a Banach space (that is, its canonical norm-induced metric is complete) if and only if it is complete as a topological vector space.
Any vector space endowed with the trivial topology (also called the indiscrete topology) is a locally convex TVS (and of course, it is the coarsest such topology). This topology is Hausdorff if and only = {}. The indiscrete topology makes any vector space into a complete pseudometrizable locally convex TVS.