定义 线性算子空间上的范数

自然语言

给定赋范线性空间 $\mathscr{X} = (X,\mathbb{R},\lVert{{\cdot}}\rVert_\mathscr{X})$ 和赋范线性空间 $\mathscr{Y} = (Y,\mathbb{R},\lVert{{\cdot}}\rVert_\mathscr{Y})$
对 $T \in \mathtip{\mathscr{L}(\mathscr{X},\mathscr{Y})}{从 \mathscr{X} 到 \mathscr{Y} 的全体有界线性算子}$ ，定义其范数为

$$ \bbox[5pt]{ \begin{aligned} \underset{\mathscr{L}(\mathscr{X},\mathscr{Y})}{\lVert{T}\rVert} & \triangleq \sup_{x \in X\backslash\theta}{\frac{\lVert{{Tx}}\rVert_\mathscr{Y}}{\lVert{{x}}\rVert_\mathscr{X}}} \\ & = \sup_{\lVert{{x}}\rVert_\mathscr{X}=1}{\lVert{{Tx}}\rVert_\mathscr{Y}} \end{aligned} } $$