Bounded linear operator between normed spaces

Definition

Let $X$ and $Y$ be normed vector spaces over the same scalar field. A linear map $T:X\to Y$ is a bounded linear operator if there is a constant $C\geq0$ such that

\lVert Tx\rVert_Y\leq C\lVert x\rVert_X \qquad\text{for every }x\in X.

The least such constant is the operator norm

\lVert T\rVert=\sup_{\lVert x\rVert_X\leq1}\lVert Tx\rVert_Y.

Boundedness here constrains the image of every vector uniformly; it does not mean that the set $T(X)$ is bounded.

Equivalent continuity conditions

For a linear map between normed spaces, the following are equivalent:

it is bounded;
it is continuous at $0$ ;
it is continuous at every point; and
it maps bounded subsets of $X$ to bounded subsets of $Y$ .

Thus bounded linear operators are precisely the normed-space instances of continuous linear maps. The equivalence uses linearity; an arbitrary continuous nonlinear map need not satisfy a global estimate by $\lVert x\rVert$ .

Operator spaces

The bounded operators from $X$ to $Y$ form a normed vector space $B(X,Y)$ under the operator norm. If $Y$ is a Banach space, then $B(X,Y)$ is Banach, whether or not $X$ is complete. When $X=Y$ , composition satisfies

\lVert ST\rVert\leq\lVert S\rVert\lVert T\rVert,

so $B(X)=B(X,X)$ is a normed algebra and is a Banach algebra when $X$ is Banach.

Examples and scope

Every linear map between finite-dimensional normed spaces is bounded. The derivative $D:C^1([0,1])\to C([0,1])$ is bounded when the source carries the $C^1$ -norm, but it is not bounded when both spaces are given the supremum norm. On an infinite-dimensional normed space, discontinuous linear maps can exist; they are everywhere-defined algebraic operators but are not bounded linear operators.

References

John B. Conway, A Course in Functional Analysis, 2nd ed., Graduate Texts in Mathematics 96, Springer, 1990. DOI record. Relevant: Chapter II on normed spaces and bounded operators.
Walter Rudin, Functional Analysis, 2nd ed., McGraw–Hill, 1991. WorldCat record. Relevant: Chapter 4 on continuous linear mappings.