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I am trying to get a hold on Hilbert Spaces, but I am having difficulties combinging various definitions.

I have looked it up on wikipedia and wolfram, there it states something like "A Hilbert space $H$ is a real or complex inner product space that is also a complete metric space with respect to the distance function induced by the inner product.".

In Croom's Principles of Topology it says "Hilbert space $H$ consists of all infinite sequences $x=(x_1,x_2,\dots,x_n,\dots )$ for which each coordinate $x_i$ is a real number and for which the sum of all $x_{i}^{2}$ converges to a finite limit $\dots$

(Im a little confused) Simply put... What is a Hilbert Space?

I am trying to get a Topologists point of view on the matter. I read that $H$ has some useful properties that come in handy when proving some theorema, so..

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Croom talks about $\ell^2(\mathbb{R})$, a Hilbert space indeed, but not the general definition. It is true, nevertheless that every infinite-dimensional separable real Hilbert space is isomorphic to $\ell^2(\mathbb{R})$. –  1015 Feb 2 '13 at 16:38
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@julien. Aha, so a Hilbert space need not to be infinite? How can i get more info on $\ell^{2}(\mathbb{R})$? What is this space called? –  omar Feb 2 '13 at 17:14
    
en.wikipedia.org/wiki/Sequence_space –  goblin Feb 2 '13 at 17:16
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@omar No, it need not be infinite-dimensional, $\mathbb{R}^n$ and $\mathbb{C}^n$ are Hilbert spaces. As you can see in user18921's comment, $\ell^2$ is called a sequence space. –  1015 Feb 2 '13 at 17:53
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Do you understand the concept of a linear (vector) space?

If so, then think of equipping the space with an inner product $\langle \cdot,\cdot\rangle$ which you can think of as a device for measuring angles in this space.

Now, that inner product induces a norm (given by $\|x\|=\langle x,x\rangle^{1/2}$) and this gives us a device for measuring length in this space.

Finally, suppose the space has the property that it is complete in the norm above, i.e. every Cauchy sequence in the space (again where distance is measured in terms of $\|\cdot\|$) has a limit which is in the space.

If all of the above is satisfied, then we say the space is a Hilbert space.

In summary, a Hilbert space is a normed, linear space with an inner product which is complete in the norm induced by the inner product.

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Consider a vector space over $\mathbb{R}$ (or $\mathbb{C}$). This can be made into an inner product space by specifying an inner product $\langle*,*\rangle,$ which takes two vectors and returns an element of $\mathbb{R}$ (or $\mathbb{C}$). Finally, a Hilbert space is a particularly nice kind of inner product space. Technically, a Hilbert Space is an inner product space that is complete with respect to the norm $ \|x\| = \langle x,x\rangle^{1/2}$. This just means it needs to be complete with respect to the metric $d(x,y) = \langle x-y,x-y\rangle^{1/2}$. Note that we have defined the metric such that $d(x,y)=\|x-y\|.$

I don't know the details about the useful properties of Hilbert space, but broadly speaking, everyone agrees that Hilbert Space is the best. Everyone loves Hilbert space.

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