从 Semi-Norms(准范数) 到 Norms(范数)

本文深入探讨了准范数(Semi-Norms)和范数(Norms)的区别,以及如何将准范数转化为范数。通过定义和实例解释了两者的性质,指出准范数在某些情况下无法满足唯一性,而范数则更为严格。通过构造商空间(Quotient Space),可以将满足特定条件的准范数空间转化为赋范空间,实现准范数到范数的转换。文章适合对线性代数和泛函分析感兴趣的读者。

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“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 VVV be a linear space over K=RK=\mathbb RK=R or over K=CK=\mathbb CK=C. A semi-norm on VVV is a function associating with every vector x∈V\mathbf x\in VxV a non-negative real number p(x)p(\mathbf x)p(x), the norm of x\mathbf xx, for which the following conditions hold:

  1. p(αx)=∣α∣p(x)p(\alpha \mathbf x)=\vert\alpha\vert p(\mathbf x)p(αx)=αp(x)
  2. p(x+y)≤p(x)+p(y)p(\mathbf x+\mathbf y)\le p(\mathbf x)+p(\mathbf y)p(x+y)p(x)+p(y)

If, moreover,

  1. p(x)=0p(\mathbf x) = 0p(x)=0 implies x=0\mathbf x = 0x=0, for all x∈V\mathbf x \in VxV,

then the function is called a norm on VVV, in which case we write ∥x∥=p(x)\Vert \mathbf x \Vert = p(\mathbf x)x=p(x).
A linear space VVV together with a semi-norm on it is called a semi-normed space, and a linear space VVV together with a norm on it is called a normed space. It is common to refer to a (semi-)normed space VVV, leaving p(x)p(\mathbf x)p(x), or ∥x∥\Vert \mathbf x \Vertx, 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 VVV是线性空间,则它必满足以下条件:

  1. 0∈V0\in V0V
  2. ∀x∈V,α∈K⇒αx∈V\forall \mathbf x\in V, \alpha\in K \Rightarrow \alpha\mathbf x\in VxV,αKαxV, where K=RK=\mathbb RK=R or K=CK=\mathbb CK=C
  3. ∀x,y∈V⇒x+y∈V\forall \mathbf x,\mathbf y\in V \Rightarrow \mathbf x+\mathbf y\in Vx,yVx+yV

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 VVV supports the trivial semi-norm given by p(x)=0p(\mathbf x) = 0p(x)=0 for all x∈V\mathbf x \in VxV. Less trivially, consider the linear space C(R,R)C(\mathbb R, \mathbb R)C(R,R) of all continuous functions x:R→Rx : \mathbb R → \mathbb Rx:RR. The evaluation semi-norm is given by ∥x∥=∣x(0)∣\Vert x\Vert = |x(0)|x=x(0).
[简译]
任意线性空间 V 都支撑由零函数(p(x)=0p(\mathbf x) = 0p(x)=0 for all x∈V\mathbf x \in VxV)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(R,R)C(\mathbb R, \mathbb R)C(R,R) of all continuous functions x:R→Rx : \mathbb R → \mathbb Rx:RR,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:∥x∥=∣x(0)∣\Vert x\Vert = |x(0)|x=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
∥αx∥=∣αx(0)∣=∣α∣∣x(0)∣=∣α∣∥x∥∥αx∥ = |αx(0)| = |α||x(0)| = |α|∥x∥αx=αx(0)=αx(0)=αx
∥x+y∥=∣x(0)+y(0)∣≤∣x(0)∣+∣y(0)∣=∥x∥+∥y∥∥x + y∥ = |x(0) + y(0)| ≤ |x(0)| + |y(0)| = ∥x∥ + ∥y∥x+y=x(0)+y(0)x(0)+y(0)=x+y.
Note that it is obvious that ∥x∥=0∥x∥ = 0x=0 need not imply x=0x = 0x=0, only that x(0)=0x(0) = 0x(0)=0, so this semi-norm is not a norm.
[简译]
实际上, ∣x(0)∣|x(0)|x(0) 满足semi-norm条件要求,是一个semi-norm。但 ∥x∥=0∥x∥ = 0x=0 仅仅指出的是 x(0)=0x(0) = 0x(0)=0,而这并不意味着 x=0x = 0x=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)x(0)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=0x = 0x=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],R)C([a, b], \mathbb R)C([a,b],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∞L_∞L or LpL_pLp norms on C([a,b],R)C([a, b], \mathbb R)C([a,b],R), then one encounters the following difficulty. It is well-known that for a non-continuous function xxx, it is possible that ∫dt⋅∣x(t)∣=0\int dt\cdot|x(t)|=0dtx(t)=0, while

### 矩阵范数与向量范数相容性的证明 #### 方法概述 为了验证矩阵范数 $\|\cdot\|$ 和向量范数 $||\cdot||$ 的相容性,需满足对于任意矩阵 $A \in R^{m\times n}$ 及其对应的任意向量 $x \in R^n$ ,有不等式成立: $$ \|Ax\|_v \leqslant \|A\|_{M} \|x\|_v $$ 其中,$\|A\|_{M}$ 表示矩阵的某种特定范数;而 $\|x\|_v,\|Ax\|_v$ 则分别表示作用前后向量所采用的一致类型的向量范数。 #### 理论依据 定义一种由给定向量范数导出的算子范数(也称为诱导范数),即设有一个已知的向量范数,则可以按照如下方式来构建相应的矩阵范数[^1]: $$ \|A\|=\sup _{\|x\|=1}\|A x\| $$ 这意味着当考虑单位球面上所有的可能输入时,该表达式的最大值就是这个新创建出来的矩阵范数。这种构造方法自然保证了上述提到的相容条件自动得以实现。 具体来说,在实际操作过程中可以通过寻找使得分母最小化的同时分子最大化的方式来求解上确界问题。例如针对 Frobenius 范数而言,它本身就是通过平方和开根号的方式得出的结果,因此可以直接利用 Cauchy-Schwarz 不等式来进行严格推导并最终确认两者间的兼容关系[^2]。 另外值得注意的是,并不是每一对随机选取的矩阵范数同向量范数组合都能保持这样的性质。只有那些遵循一定规律建立起来的关系才具有此特性,比如前面提及到的一些经典例子:向量 1-范数对应于矩阵列和的最大者作为 M1 范数;向量无穷大范数关联着行和中的最大绝对值得到 Minf 范数等等[^3]。 ```python import numpy as np def check_compatibility(A, vector_norm_func, matrix_norm_func): """ Check compatibility between a given matrix norm and vector norm. Parameters: A (numpy.ndarray): The input matrix to test on. vector_norm_func (callable): Function that computes the vector norm of an array-like object. matrix_norm_func (callable): Function that computes the induced matrix norm from `vector_norm_func`. Returns: bool: Whether or not they are compatible according to definition provided earlier. """ max_ratio = -np.inf # Generate random vectors within unit sphere under specified vector norm for i in range(1000): v = np.random.randn(len(A.T)) normalized_v = v / vector_norm_func(v) ratio = vector_norm_func(np.dot(A, normalized_v)) / vector_norm_func(normalized_v) if ratio > max_ratio: max_ratio = ratio return abs(max_ratio - matrix_norm_func(A)) < 1e-8 # Example usage with L2 norms both for vectors and matrices print(check_compatibility( np.array([[1., 2.], [3., 4.]]), lambda x : np.linalg.norm(x), lambda A : np.linalg.norm(A, ord='fro') )) ```
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