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In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X → Y {\displaystyle L:X\to Y} between topological vector spaces X {\displaystyle X} and Y {\displaystyle Y} that maps bounded subsets of X {\displaystyle X} to bounded subsets of Y . {\displaystyle Y.} If X {\displaystyle X} and Y {\displaystyle Y} are normed vector spaces , then L {\displaystyle L} is bounded if and only if there exists some M > 0 {\displaystyle M>0} such that for all x ∈ X , {\displaystyle x\in X,}
The concept of a bounded linear operator has been extended from normed spaces to certain to all topological vector spaces.
Outside of functional analysis, when a function f : X → Y {\displaystyle f:X\to Y} is called "bounded" then this usually means that its image f {\displaystyle f} is a bounded subset of its codomain. A linear map has this property if and only if it is identically 0. {\displaystyle 0.} Consequently, in functional analysis, when a linear operator is called "bounded" then it is never meant in this abstract sense.