Bounded Linear Functional and Its Norm

Let $X$ be a normed space over $\mathbb K\in\{\mathbb R,\mathbb C\}$ . A linear map $f:X\to\mathbb K$ is a linear functional. It is bounded if there exists $M\ge0$ such that

|f(x)|\le M\lVert x\rVert\qquad\text{for every }x\in X.

Its operator norm is

\lVert f\rVert=\sup_{\lVert x\rVert\le1}|f(x)|.

Equivalent characterizations

Equivalently,

\lVert f\rVert=\inf\{M\ge0:|f(x)|\le M\lVert x\rVert\text{ for every }x\in X\}.

Boundedness is equivalent to continuity.

Remarks

This notion is used in Hahn–Banach in normed spaces and in separation results such as separating a point and a subspace.

Examples

On $X=\mathbb R^n$ with the Euclidean norm, $f(x)=\langle a,x\rangle$ has norm $\lVert a\rVert_2$ .
On $C[0,1]$ with the supremum norm, evaluation $f(x)=x(t_0)$ has norm $1$ .