Implications of inner products vs normed spaces vs metric spaces

Question

Is it true that:

-an inner product satisfies the properties of a norm if and only if the norm satisfies the parallelogram equality

-a norm can be induced by a metric if and only if the metric satisfies $d(x+a,y+a)=d(x,y)$ and $d(ax,ay)=ad(x,y)$

or are the implications one way?

The wording seems to be a bit confused: The first statement should be something like, "a (vector space) norm is induced by an inner product iff it satisfies the parallelogram identity." In the second, a norm is a unary map $X \to \mathbb{R}$ but the notation here even indicates that a metric is a binary map $X \times X \to \mathbb{R}$, so a norm is never a metric---to ask a sensible question, we must first describe how to build a candidate metric in terms of a norm and/or vice versa. — Travis Willse, Apr 20 '15 at 10:09
sorry yes my wording for the 2nd part was poor. What i meant was a norm can be induced by a metric by setting ||x-y||=d(x,y) iff ... — Dan Burrows, Apr 20 '15 at 10:15
And I'm thinking that these are indeed iff statements, though I've never seen them described this way, hence the question. — Dan Burrows, Apr 20 '15 at 10:18
Your inner product lines still have the problem that Travis pointed out. $;$ — , Apr 20 '15 at 10:36
Not every metric is induced from a norm, When a metric space is a normed space?, Was this metric induced by a norm? — Martin Sleziak, Apr 20 '15 at 11:17

score 1 · Accepted Answer · answered Aug 29 '15 at 18:27

The first statement is true both ways. Specifically, suppose $(X, ||\cdot||)$ is a normed linear space. Then the norm $||\cdot ||$ is induced by an inner product iff the parallelogram law holds in $(X,||\cdot||)$.

For the second statement, this is not true. Call the condition $d(x,y)=d(x+a,y+a)$ translation invariance, and the condition $d(x,y)=d(ax,ay)$ homogeneity. Consider the following variant of the characteristic function $$ \chi(x,y)=\left\{\begin{matrix} 1, & x\neq y, \\ 0, & \text{otherwise.} \end{matrix}\right. $$ Homogeneity fails for $\chi,~x\neq y$. Indeed, $$ \chi(x,y)=1\neq a=\chi(ax,ay). $$

Implications of inner products vs normed spaces vs metric spaces

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