# Every inner product space is a normed space which is also a metric space

As I am pretty sure that everybody knows that a Hilbert space is a space that is a complete, separable and generally infinite dimensional inner product space. By the means of completion, every Cauchy sequence is convergent in this space. Besides, every inner product space is a normed space which is also a metric space, that can be shown as folows:

$d(\vec{x}, \vec{y})=\| \vec{x}-\vec{y}\|$

However, in a document online, I just saw this notation below which I can not understand.

$d(\vec{x}, \vec{y})=\| \vec{x}-\vec{y}\| =\sqrt{\langle \vec{x}, \vec{y}\rangle}$

How can the third part of this equality above hold? Thanks in advance.

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I believe that should be $\langle \vec{x} - \vec{y}, \vec{x} - \vec{y} \rangle$. –  Mark Fantini Jan 12 '14 at 11:04
The document online contains a big misprint. –  Siminore Jan 12 '14 at 11:05
But it is a Master thesis from Canada. Fantini, yeah right, I was just going to edit my original post that it should be like that! As far as I am concerned about that it is a thesis, plus I am working on Computer Science, I was a bit skeptical about the answer. –  Oceansoul Jan 12 '14 at 11:08
@StefanSmith Oh, That's my mistake. Right, Euclidean space is finite example for Hilbert spaces. I will edit it then, because there are some infinite dimensional Hilbert spaces. Besides, it is no reference for me. I am just working on Kernel methods and its algebric background. That document is what appeals to me most, you know, easy to understand et cetera. anyway, much obliged for your precious comments. –  Oceansoul Jan 12 '14 at 21:01