So this is how I went about this: $\langle\,\cdot\,,\,\cdot\,\rangle: X \times X \to \mathbb{R}$ such that (by definition I list the properties of scalar product) and I can east prove the first three properties of the metric defined as $$d(x,y)= \sqrt{\langle x-y,x-y\rangle}$$ but I am having trouble with the triangle inequality, I tried adding and subtracting z in the scalar product , with hopes of somehow sorting out the inequality using the properties of scalar product , but to no luck..
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You have that
$$d(x,z)^2 = \langle x-z, x-z \rangle = \langle x-y+y-z, x-y+y-z \rangle$$
$$= \langle x-y, x-y \rangle + 2 \langle x-y, y-z \rangle + \langle y-z, y-z \rangle$$
$$ = d(x,y)^2 + 2 \langle x-y, y-z \rangle + d(y,z)^2$$
Then by Cauchy-Schwarz,
$$ \leq d(x,y)^2 + 2 d(x,y)d(y,z) + d(y,z)^2$$
$$ \leq \left( d(x,y) + d(y,z) \right)^2$$
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<and>mean "less than" and "greater than", and produce spacing correct for that meaning only; to make angle brackets, use\langleand\rangle. – Zev Chonoles Jul 07 '15 at 21:07