In functional analysis and operator theory, a bounded linear operator is a linear transformation between topological vector spaces (TVSs) and that maps bounded subsets of to bounded subsets of
If and are normed vector spaces (a special type of TVS), then is bounded if and only if there exists some such that for all
The smallest such is called the operator norm of and denoted by
A bounded operator between normed spaces is continuous and vice versa.
The concept of a bounded linear operator has been extended from normed spaces to all topological vector spaces.
Outside of functional analysis, when a function is called "bounded" then this usually means that its is a bounded subset of its codomain. A linear map has this property if and only if it is identically
Consequently, in functional analysis, when a linear operator is called "bounded" then it is never meant in this abstract sense (of having a bounded image).
Every bounded operator is Lipschitz continuous at
A linear operator between normed spaces is bounded if and only if it is continuous.
A linear operator between two topological vector spaces (TVSs) is called a or just if whenever is bounded in then is bounded in
A subset of a TVS is called bounded (or more precisely, von Neumann bounded) if every neighborhood of the origin absorbs it.
In a normed space (and even in a seminormed space), a subset is von Neumann bounded if and only if it is norm bounded.
Hence, for normed spaces, the notion of a von Neumann bounded set is identical to the usual notion of a norm-bounded subset.
Every sequentially continuous linear operator between TVS is a bounded operator.
This implies that every continuous linear operator between metrizable TVS is bounded.
However, in general, a bounded linear operator between two TVSs need not be continuous.
This formulation allows one to define bounded operators between general topological vector spaces as an operator which takes bounded sets to bounded sets.
In this context, it is still true that every continuous map is bounded, however the converse fails; a bounded operator need not be continuous.
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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. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family.
In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions f defined on a set S the non-negative number This norm is also called the , the , the , or, when the supremum is in fact the maximum, the . The name "uniform norm" derives from the fact that a sequence of functions \left{f_n\right} converges to f under the metric derived from the uniform norm if and only if f_n converges to f uniformly.
In functional analysis and related areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector can be inflated to include the set. A set that is not bounded is called unbounded. Bounded sets are a natural way to define locally convex polar topologies on the vector spaces in a dual pair, as the polar set of a bounded set is an absolutely convex and absorbing set. The concept was first introduced by John von Neumann and Andrey Kolmogorov in 1935.
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