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Norm Applied To A Linear Map

TOPO-JFEQB1

Suppose that $V,W$ are finite dimensional linear spaces, and that $A:V\to W$ is a linear map from $V$ to $W$.

A non-negative map $v\mapsto N(v)$ is called a seminorm if $N(av)= |a| N(v)$ for every $a$ real number and every vector $v$ and if $N(u+v)\leq N(u)+N(v)$ for any two vectors $u,v$.

Let $\|\cdot \|_W$ be a norm on $W$, and define

$$\|x\| := \|Ax\|_W.$$

Then it is necessarily TRUE that:

A

$x \mapsto \|x\|$ is non-negative.

B

$\|ax\|= |a|\cdot \|x\|$ for all $x\in V$ and all $a\in \mathbb{R}$.

C

$\|x\|=0$ if and only if $x$ is the zero-vector.

D

$\|x+y\|\leq \|x\|+\|y\|$ for all $x,y\in V$.

E

$x\maps\to \|x\|$ is a seminorm on $V$.

F

$(x,y)\mapsto d(x,y):=\|x-y\|$ is a metric on $V$.