Norms

Related: Metrics, metric spaces, and the metric topology

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Norms on inner product spaces §

Definition: Norm on an inner product space

If $V$ is an Inner product space, the norm, or magnitude, is a function $\Vert \cdot \Vert :V\to [0,\infty )$ defined by $\sqrt{⟨v,v⟩}\in [0,\infty )$.

Abstract MATH-UN1207, Lemma 102

The norm on an inner product space satisfies the following properties:

• (N1) $\Vert v\Vert \ge 0$, and $\Vert v\Vert =0⟺v=\vec{0}$.
• (N2) $\Vert cv\Vert =|c|\Vert v\Vert$
• (N3) $\Vert v+w{\Vert }^2=\Vert v{\Vert }^2+\Vert w{\Vert }^2+2\text{Re}⟨v,w⟩$. In particular, $\Vert v+w{\Vert }^2=\Vert v{\Vert }^2+\Vert w{\Vert }^2$ if and only if $⟨v,w⟩$ has no real part.

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Norms on ${\mathbb{R}}^n$ §

Definition: Norm on \mathbb R^n

A norm on ${\mathbb{R}}^n$ is a function $N:{\mathbb{R}}^n\to [0,\infty )$ which satisfies the following properties:

• $N(\vec{x})=0⟺\vec{x}=\vec{0}$
• $N(c\vec{x})=|c|N(\vec{x})$
• $N(\vec{x}+\vec{y})\le N(\vec{x})+N(\vec{y})$

The $p$-norm function is defined as

$$N_p\left(\begin{array}{c}{x}_1\\ \cdots \\ x_n\end{array}\right)=[|x_1{|}^p+\cdots +|x_n{|}^p{]}^{1/p}$$

for $1\le p<\infty$, and

$$N_{\infty }\left(\begin{array}{c}{x}_1\\ \cdots \\ x_n\end{array}\right)=\text{max}(|x_1|,\cdots ,|x_n|)$$

for $p=\infty$.

For example, the Euclidian norm or $2$-norm is given by

$$\Vert (x_1\cdots x_n{)}^T{\Vert }_2=\sqrt{x_1^2+\cdots +x_n^2}.$$

Definition: Equivalent norms

Two norms $N,N^{\prime }$ are equivalent if there exist positive constants $c,C>0$ so that

$$c{N}^{\prime }(\vec{x})\le N(\vec{x})\le C{N}^{\prime }(\vec{x}).$$