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As mentioned in title, are the any connections between inner product space(well, here we talk about only real space) and metric space? I kind of notice that the axioms satisfied by both inner product and metric are almost the same.

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Yes, every inner product space is a metric space, with the "Euclidean metric" defined by $$ d(x,y) = \sqrt{\langle x-y, x-y \rangle} $$

Not every metric on a vector space comes from an inner product though (For instance, $l^1$, the space of summable sequences, is one such example)

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  • $\begingroup$ So what you mean is inner product space is metric space but not the vice versa? $\endgroup$
    – Idonknow
    Aug 13, 2013 at 16:47
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    $\begingroup$ Please read my answer below: a metric space can just be a finite set $\endgroup$
    – Avitus
    Aug 13, 2013 at 16:47
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There is even an intermediate step: in your setting one can prove that

$$\text{inner product}\Rightarrow \text{norm}\Rightarrow \text{metric/distance},$$

where $\Rightarrow$ means "induces". A first, major difference between the above structures is that inner products and norms require a vector space structure, while metrics are defined on sets.

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