How do I prove that the inner product operator is a metric?

Solution 1:

Your definition of inner product is wrong. Your first and last properties are contradictory: $$ 0\leq \langle v,w\rangle=-\langle -v,w\rangle\leq0, $$ so the only choice you have is $\langle v,w\rangle=0$ for all $v,w$.

Instead, in the definition of inner product what you require is that $\langle v,v\rangle\geq0$.

Your second property is also wrong, again in easy contradiction with your last property: it should be $\langle v,v\rangle=0\implies v=0$. Otherwise you miss the crucial feature of an inner product which is orthogonality.

Another important feature that you miss is that $M$ should be a vector space. Otherwise, $u+v$ makes no sense, nor would multiplication by a scalar.

The way to obtain a distance from an inner product, is by $$ d(x,y)=\sqrt{\langle x-y,x-y\rangle}. $$ This is usually phrased in terms of a norm, i.e. $$ d(x,y)=\|x-y\|, $$ where the norm is $$ \|x\|=\sqrt{\langle x,x\rangle}. $$