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I have come across the following example of a non-separable Hilbert space:

Example 2.84. Let $I$ be a set, equipped with the discrete topology and the counting measure $\lambda_{\text{ count}}$ defined on the $\sigma$-algebra $\Bbb P(I)$ of all subsets of $I$. Then $$\ell^2(I)=L^2\big(I,\Bbb P(I),\lambda_{\text{ count}}\big)$$ is a Hilbert space, and it comprises all functions $a:I\to\Bbb R$ (or $\Bbb C$) for which the support $$F=\{i\in I : a(i)\ne0\},$$ is finite or countable, and for which $\sum_{i\in I}|a_i|^2=\sum_{i\in F}|a_i|^2\lt\infty$.

Why do I need the discrete topology on $I$? Or more generally: why do I need a topology? If we talk about $L^p$ spaces in general, we only want a measure space and we don't mention a topology because $f \in L^p$ doesn't have to be continuous. Thanks for your help.

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    We do not need any topology here2012-08-02
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    Maybe topology is useful in an other part of the example (if the whole example is displayed, I don't see). A remark: the Hilbert space we define in such a way can be separable (when $I$ is finite or countable), otherwise it isn't.2012-08-02
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    @DavideGiraudo Yes it is the whole example, after where I cut, a new chapter starts. Many thanks for your remark!2012-08-02
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    Maybe earlier examples were about $L^p$-spaces with respect to Radon measures on locally compact spaces? The discrete topology is locally compact and $\mathbb P(I)$ is the Borel $\sigma$-algebra, etc. hence compactly supported (=finitely supported) functions are dense.2012-08-02
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    @DavideGiraudo Perhaps they wanted to write "Let $I$ be an uncountable set..."2012-08-02
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    @t.b. The last example before this one is an entire chapter entitled "Radon-Nikodym derivative" which I skipped. But in between there are two other (mini-)chapters. But even so: I think one can safely consider this a typo.2012-08-02
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    I don't see a typo here. It is in the mindset of many people to equip a set with the discrete topology by reflex when it has none and to say that explicitly. That's why you find "Let $G$ be a discrete group" all over the place even when there's no topological group whatsoever in sight.2012-08-02
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    Hm. By now I think that we also don't need a measure on that space. For an example of a non-separable Hilbert space take $I$ uncountable, then $\ell^2 (I)$ is a non-separable Hilbert space if endowed with the inner-product $\langle x,y \rangle = \frac{1}{4}(\|x + y\|_2^2 - \|x-y\|_2^2)$.2012-08-04
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    I think it's possible to construct the space in the way you suggest in your last comment. But this way - with the counting measure - the author of that text is able to do it as a special case of $L_2$, which he had defined before. (BTW what text is this from?)2012-08-04
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    @MartinSleziak Course notes by Einsiedler/Ward. Would you like to have a link to it? : )2012-08-04
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    I have downloaded it from here: http://www.math.ethz.ch/~einsiedl/FA-lecture.pdf But it seems to be a newer version - this one has it as Example 2.88. (It writes "Draft July 2, 2012" in the title.) I think that if your question is based on some book or lecture notes, it would be nice (for the benefit of other users) to mention it in the post. (And add a link, if it is available.)2012-08-04

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