Bounded operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation formula_1 between topological vector spaces (TVSs) formula_2 and formula_3 that maps bounded subsets of formula_2 to bounded subsets of formula_5 If formula_2 and formula_3 are normed vector spaces (a special type of TVS), then formula_8 is bounded if and only if there exists some formula_9 such that for all formula_10 formula_11 The smallest such formula_12 is called the operator norm of formula_8 and denoted by formula_14 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 certain to all topological vector spaces. Outside of functional analysis, when a function formula_15 is called "bounded" then this usually means that its image formula_16 is a bounded subset of its codomain. A linear map has this property if and only if it is identically formula_17 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). In normed vector spaces. Every bounded operator is Lipschitz continuous at formula_17 Equivalence of boundedness and continuity. A linear operator between normed  spaces is bounded if and only if it is continuous. In topological vector spaces. A linear operator formula_19 between two topological vector spaces (TVSs) is called a ' or just ' if whenever formula_20 is bounded in formula_2 then formula_22 is bounded in formula_5 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. Continuity and boundedness. Every sequentially continuous linear operator between TVS is a bounded operator. This implies that every continuous linear operator 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. This also means that boundedness  is no longer equivalent to Lipschitz continuity in this context. If the domain is a bornological space (for example, a pseudometrizable TVS, a Fréchet space, a normed space) then a linear operators into any other locally convex spaces is bounded if and only if it is continuous. For LF spaces, a weaker converse holds; any bounded linear map from an LF space is sequentially continuous. If formula_19 is a linear operator between two topological vector spaces and if there exists a neighborhood formula_25 of the origin in formula_2 such that formula_27 is a bounded subset of formula_28 then formula_29 is  continuous. This fact is often summarized by saying that a linear operator that is bounded on some neighborhood of the origin is necessarily continuous. In particular, any linear functional that is bounded on some neighborhood of the origin is continuous (even if its domain is not a normed space). Bornological spaces. Bornological spaces are exactly those locally convex spaces for which every bounded linear operator into another locally convex space is necessarily continuous. That is, a locally convex TVS formula_2 is a bornological space if and only if for every locally convex TVS formula_28 a linear operator formula_19 is continuous  if and only if it is bounded. Every normed space is bornological. Characterizations of bounded linear operators. Let formula_19 be a linear operator between topological vector spaces (not necessarily Hausdorff). The following are equivalent: if formula_2 and formula_3 are locally convex then the following may be add to this list: if formula_2 is a bornological space and formula_3 is locally convex then the following may be added to this list: Examples. Unbounded linear operators. Let formula_2 be the space of all trigonometric polynomials on formula_51 with the norm formula_52 The operator formula_53 that maps a polynomial to its derivative is  not bounded. Indeed, for formula_54 with formula_55 we have formula_56 while formula_57 as formula_58 so formula_8 is not bounded.