Norm of an operator

• categories: Linear algebra, Definition

Equivalent Definitions of the Operator Norm

The operator norm of a Linear map $T:X\to Y$, where $X$ and $Y$ are normed vector spaces, measures the “size” of $T$ in terms of how much it can stretch vectors. The operator norm is defined by several equivalent formulations:

Definition 1: Supremum Form

$$\Vert T\Vert =\underset{\Vert x\Vert \le 1}{\sup }\Vert T(x)\Vert$$

This defines $\Vert T\Vert$ as the supremum of $\Vert T(x)\Vert$ over all $x$ with $\Vert x\Vert \le 1$, meaning it is the maximum amount $T$ can stretch any vector of unit length.

Definition 2: Infimum of Bounds

$$\Vert T\Vert =\inf \{C\ge 0:\Vert T(x)\Vert \le C\Vert x\Vert ,\forall x\in X\}$$

This characterizes $\Vert T\Vert$ as the smallest constant $C$ such that $T$ is bounded by $C$ times the norm of $x$, for all $x\in X$.

Definition 3: Supremum of Ratios (if $x\ne 0$)

$$\Vert T\Vert =\underset{x\ne 0}{\sup }\frac{\Vert T(x)\Vert }{\Vert x\Vert }$$

This expresses $\Vert T\Vert$ as the supremum of the ratio $\frac{\Vert T(x)\Vert }{\Vert x\Vert }$ over all non-zero $x$, capturing the maximal relative stretching of $T$.

Properties

• Non-negativity: $\Vert T\Vert \ge 0$, and $\Vert T\Vert =0$ if and only if $T$ is the zero map.
• Sub-multiplicativity: If $T:X\to Y$ and $S:Y\to Z$ are operators, then $\Vert ST\Vert \le \Vert S\Vert \cdot \Vert T\Vert$.
• Triangle Inequality: $\Vert T+S\Vert \le \Vert T\Vert +\Vert S\Vert$.

Each of these definitions provides a different perspective but leads to the same norm value for the operator $T$.