Bounded operator

Article summary

In functional analysis and operator theory, a bounded linear operator is a special kind of linear transformation that is particularly important in infinite dimensions. In finite dimensions, a linear transformation takes a bounded set to another bounded set (for example, a rectangle in the plane goes either to a parallelogram or bounded line segment when a linear transformation is applied). However, in infinite dimensions, linearity is not enough to ensure that bounded sets remain bounded: a bounded linear operator is thus a linear transformation that sends bounded sets to bounded sets. Formally, it is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector spaces (a special type of TVS), then L is bounded if and only if there exists some M \ge 0 such that for all x \in X, \|Lx\|_Y \leq M \|x\|_X. The infimum such M is called the operator norm of L and denoted by \|L\|. A linear operator between normed spaces is continuous if and only if it is bounded. 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 f : X \to Y is called "bounded" then this usually means that its image f(X) is a bounded subset of its codomain.…

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