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mathematical analysis > functional analysis > operator > bounded operator

algebra > linear algebra > linear map > bounded operator

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bounded operator

In functional analysis and operator theory, a bounded linear operator is a linear transformation ${\\displaystyle L:X\ o Y}$ between topological vector spaces (TVSs) ${\\displaystyle X}$ and ${\\displaystyle Y}$ that maps bounded subsets of ${\\displaystyle X}$ to bounded subsets of ${\\displaystyle Y.}$
If ${\\displaystyle X}$ and ${\\displaystyle Y}$ are normed vector spaces (a special type of TVS), then ${\\displaystyle L}$ is bounded if and only if there exists some ${\\displaystyle M>0}$ such that for all ${\\displaystyle x\\in X,}$
${\\displaystyle \\|Lx\\|_{Y}\\leq M\\|x\\|_{X}.}$
The smallest such ${\\displaystyle M}$ is called the operator norm of ${\\displaystyle L}$ and denoted by ${\\displaystyle \\|L\\|.}$
A bounded operator between normed spaces is continuous and vice versa.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Bounded_operator)

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