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Concept# Bounded operator

Summary

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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