Bounded Linear Functional and Its Norm

A linear functional is bounded iff it is continuous; its operator norm is sup_{||x||≤1}|f(x)|.

Bounded Linear Functional and Its Norm

Let $X$ be a normed space over $\mathbb{R}$ or $\mathbb{C}$ . A linear map $f:X\to\mathbb{K}$ (where $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$ ) is a linear functional.

The functional $f$ is bounded if there exists $M>0$ such that

|f(x)|\le M\|x\|\quad\text{for all }x\in X.

In normed spaces, boundedness is equivalent to continuity.

The norm (operator norm) of $f$ is

\|f\|:=\sup_{\|x\|\le 1}|f(x)|.

Equivalently, $\|f\|=\inf\{M>0: |f(x)|\le M\|x\|\ \forall x\}$ .

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, $f(x)=\langle a,x\rangle$ is bounded and $\|f\|=\|a\|_2$ .
If $X=C[0,1]$ with $\|\cdot\|_\infty$ , then $f(x)=x(t_0)$ is bounded with $\|f\|=1$ .