从 Semi-Norms（准范数） 到 Norms（范数）

“什么是 Semi-Norms 什么是 Norms？它们之间有什么关系，为什么有了Norm还要有Semi-Norm呢？它们之间如何转换呢？”
以上是我在学习Semi-Norms（准范数）和 Norms（范数）时心中一直萦绕的问题，而且很久了，许多书都没讲清楚，直到我看到了【1】。在该书的第5章中有一节《Semi-Norms and Norms》，将它们之间的关系非常清楚地表述了出来，此处，我就自己的一些理解和翻译作文以小结之。

一、什么是Semi-Norms（准范数）和 Norms（范数）

Definition 1.1: (Semi-Norm and Norm)
Let $V$ be a linear space over $K=\mathbb{R}$ or over $K=\mathbb{C}$. A semi-norm on $V$ is a function associating with every vector $\mathbf{x}\in V$ a non-negative real number $p(\mathbf{x})$, the norm of $\mathbf{x}$, for which the following conditions hold:

1. $p(\alpha \mathbf{x})=|\alpha |p(\mathbf{x})$
2. $p(\mathbf{x}+\mathbf{y})\le p(\mathbf{x})+p(\mathbf{y})$

If, moreover,

3. $p(\mathbf{x})=0$ implies $\mathbf{x}=0$, for all $\mathbf{x}\in V$,

then the function is called a norm on $V$, in which case we write $\Vert \mathbf{x}\Vert =p(\mathbf{x})$.
A linear space $V$ together with a semi-norm on it is called a semi-normed space, and a linear space $V$ together with a norm on it is called a normed space. It is common to refer to a (semi-)normed space $V$, leaving $p(\mathbf{x})$, or $\Vert \mathbf{x}\Vert$, implicit.
以上定义给出了准范数（Semi-Norms）和范数（Norms）的差别，即范数多了条件3，而准范数没有此要求。另外，上述定义是在线性空间中定义的，什么是线性空间（Linear Space）呢？所谓线性空间就是一个集合（Set），该集合包含零元（0），而且对矢量加和标量乘封闭，即：若 $V$是线性空间，则它必满足以下条件：

1. $0\in V$
2. $\forall \mathbf{x}\in V,\alpha \in K\Rightarrow \alpha \mathbf{x}\in V$, where $K=\mathbb{R}$ or $K=\mathbb{C}$
3. $\forall \mathbf{x},\mathbf{y}\in V\Rightarrow \mathbf{x}+\mathbf{y}\in V$

范数是在线性空间上讨论的，这点很重要。

2、二者到底有什么差别

虽然通过上述定义的条件上可以看出两者的差别，但仍然对此差别没有什么感觉，我们来看一个例子：
Firstly, and quite trivially, any linear space $V$ supports the trivial semi-norm given by $p(\mathbf{x})=0$ for all $\mathbf{x}\in V$. Less trivially, consider the linear space $C(\mathbb{R},\mathbb{R})$ of all continuous functions $x:\mathbb{R}\to \mathbb{R}$. The evaluation semi-norm is given by $\Vert x\Vert =|x(0)|$.
[简译]
任意线性空间 V 都支撑由零函数（$p(\mathbf{x})=0$ for all $\mathbf{x}\in V$）定义的准泛函，这是一个平凡的semi-norm。若要给出一个不那么平凡的semi-norm，我们考虑在R上的全体连续函数：$C(\mathbb{R},\mathbb{R})$ of all continuous functions $x:\mathbb{R}\to \mathbb{R}$，它的每个元都是一个从实数域到实数域的函数（映射），该空间的semi-norm定义为：$\Vert x\Vert =|x(0)|$，即每个元的范数等于该元（函数）在0点函数值的绝对值。

Indeed, and
$\Vert \alpha x\Vert =|\alpha x(0)|=|\alpha ||x(0)|=|\alpha |\Vert x\Vert$
$\Vert x+y\Vert =|x(0)+y(0)|\le |x(0)|+|y(0)|=\Vert x\Vert +\Vert y\Vert$.
Note that it is obvious that $\Vert x\Vert =0$ need not imply $x=0$, only that $x(0)=0$, so this semi-norm is not a norm.
[简译]
实际上， $|x(0)|$ 满足semi-norm条件要求，是一个semi-norm。但 $\Vert x\Vert =0$ 仅仅指出的是 $x(0)=0$，而这并不意味着 $x=0$，所以this semi-norm is not a norm。

This construction is a typical one giving rise to semi-norms by ignoring some of the information embodied in the vectors (in this case, only $x(0)$ is important for determining the semi-norm, all the other values of the function are ignored). Naturally, one may evaluate at any point, not just at $x=0$, and obtain a family of semi-norms.
[简译]上述的 semi-norm 只取了函数的一个点作为它的准范数，而忽略了其它点的值，由此观之，我们亦可选择其它点，并得到 a family of semi-norms.

We mention here that there is another natural way to obtain semi-norms. The space $C([a,b],\mathbb{R})$ is a space of continuous functions. If one wishes to consider non-continuous functions and to utilize the integral in order to obtain a norm similar in ?spirit to the $L_{\infty }$ or $L_p$ norms on $C([a,b],\mathbb{R})$, then one encounters the following difficulty. It is well-known that for a non-continuous function $x$, it is possible that $\int dt\cdot |x(t)|=0$, while