Kernel

Hilbert Space Definition 1 (Norm) Let $\mathcal{F}$ be a vector space over $\mathbb{R}$ (For example $\mathcal{F}=\mathbb{R}^n$ is a vector space). A function $||\cdot||_{\mathcal{F}}: \mathcal{F} \to [0, \inf)$ is said to be a norm on $\mathcal{F}$ if ($||\cdot||_{\mathcal{F}}$ 是一个有效norm算子要满足以下条件): For $f \in \mathcal{F}$, $||f||_{\mathcal{F}}=0$ if and only if $f=0$. (norm separates points) $|\lambda f|_{\mathcal{F}}=|\lambda||f|_{\mathcal{F}}$, $\forall \lambda \in \mathbb{R}, \forall f \in \mathcal{F}$ (positive homogeneity). $|f+g|_{\mathcal{F}} \leq|f|_{\mathcal{F}}+|g|_{\mathcal{F}}, \forall f, g \in \mathcal{F}$ (triangle inequality). 向$||\cdot||_{\mathcal{F}}$中输入任意一个向量，只要满足以上条件，那么$||\cdot||_{\mathcal{F}}$是一个valid norm operator. ...