In functional analysis and operator theory, a bounded linear operator 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 > 0 such that for all x \in X, The smallest such M is called the operator norm of L and denoted by |L|. 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 f : X \to Y is called "bounded" then this usually means that its image f(X) is a bounded subset of its codomain. A linear map has this property if and only if it is identically 0. 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 0.

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 F : X \to Y between two topological vector spaces (TVSs) is called a **' or just **' if whenever is bounded in X then F(B) is bounded in Y. 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 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. 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 F : X \to Y is a linear operator between two topological vector spaces and if there exists a neighborhood U of the origin in X such that F(U) is a bounded subset of Y, then F 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 X is a bornological space if and only if for every locally convex TVS Y, a linear operator F : X \to Y is continuous if and only if it is bounded. Every normed space is bornological.

Characterizations of bounded linear operators

Let F : X \to Y be a linear operator between topological vector spaces (not necessarily Hausdorff). The following are equivalent: if X and Y are locally convex then the following may be add to this list:

<li>F maps bounded [disks](https://bliptext.com/articles/absolutely-convex-set) into bounded [disks](https://bliptext.com/articles/absolutely-convex-set).</li> <li>F^{-1} maps [bornivorous](https://bliptext.com/articles/bornivorous-set) disks in Y into [bornivorous](https://bliptext.com/articles/bornivorous-set) disks in X.</li> </ol> if X is a [bornological space](https://bliptext.com/articles/bornological-space) and Y is locally convex then the following may be added to this list: <li>F is [sequentially continuous at some](https://bliptext.com/articles/sequential-continuity-at-a-point) (or equivalently, at every) point of its domain. <li>F is [sequentially continuous at the origin](https://bliptext.com/articles/sequential-continuity-at-a-point).</li> </ol>

Examples

<ul> <li>Any linear operator between two finite-dimensional normed spaces is bounded, and such an operator may be viewed as multiplication by some fixed [matrix](https://bliptext.com/articles/matrix-mathematics).</li> <li>Any linear operator defined on a finite-dimensional normed space is bounded.</li> <li>On the [sequence space](https://bliptext.com/articles/sequence-space) c_{00} of eventually zero sequences of real numbers, considered with the \ell^1 norm, the linear operator to the real numbers which returns the sum of a sequence is bounded, with operator norm 1. If the same space is considered with the norm, the same operator is not bounded.</li> <li>Many [integral transforms](https://bliptext.com/articles/integral-transform) are bounded linear operators. For instance, if is a continuous function, then the operator L defined on the space C[a, b] of continuous functions on [a, b] endowed with the [uniform norm](https://bliptext.com/articles/uniform-norm) and with values in the space C[c, d] with L given by the formula is bounded. This operator is in fact a [compact operator](https://bliptext.com/articles/compact-operator). The compact operators form an important class of bounded operators.</li> <li>The [Laplace operator](https://bliptext.com/articles/laplace-operator) (its [domain](https://bliptext.com/articles/domain-of-a-function) is a [Sobolev space](https://bliptext.com/articles/sobolev-space) and it takes values in a space of [square-integrable functions](https://bliptext.com/articles/square-integrable-function)) is bounded.</li> <li>The [shift operator](https://bliptext.com/articles/shift-operator) on the [Lp space](https://bliptext.com/articles/lp-space) \ell^2 of all [sequences](https://bliptext.com/articles/sequence) of real numbers with is bounded. Its operator norm is easily seen to be 1.</li> </ul>

Unbounded linear operators

Let X be the space of all trigonometric polynomials on with the norm The operator L : X \to X that maps a polynomial to its derivative is not bounded. Indeed, for with we have while as so L is not bounded.

Properties of the space of bounded linear operators

The space of all bounded linear operators from X to Y is denoted by B(X, Y).