In my homework for functional analysis, we had $\mathcal{H} = \ell^2(\mathbb{N})$. But the usual sequence space is defined to be a vector space over complex or real numbers. I'm not sure how $\ell^2(\mathbb{N})$ is a Hilbert space.
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It's not even a group. – Bernard Feb 02 '18 at 21:11
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This is the Hilbert space. (Others are either non-separable or isomorphic to this one.) – user334639 Feb 02 '18 at 21:46
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@user334639: there's the finite-dimensional ones :) – Martin Argerami Feb 02 '18 at 23:43
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As Wiki says, $\ell^p$ are special cases of $L^p$ where the space is $\mathbf N$ and the measure is the counting measure. So, is in fact the notation $\ell^p(\mathbf N)$ is the same as you would use for $L^p$. – Dog_69 Feb 03 '18 at 00:24
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$\ell^2(\mathbb{N})$ is the space of the (complex or real) sequences $x = (x_n)_{n\in\Bbb N}$ s.t. $$\sum_{n\in\Bbb N}|x_n|^2 < \infty.$$ The scalar product is $$(x,y) = \sum_{n\in\Bbb N}x_n\bar{y_n}.$$
Martín-Blas Pérez Pinilla
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Just clarifying some notation for myself, I can't really find it explicitly anywhere. Is $\ell^p(\mathbb{N})$ just the same as $\ell^p$? And in general $\ell^p(K)$ is sequences $(a_i)_{i\in K}$? What can $K$ be in that case, any countable set? – Christopher.L Nov 02 '19 at 20:35
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@Christopher.L, yes, yes and usually yes. In fact $K$ can be uncountable, but see the problems in https://math.stackexchange.com/questions/106102/use-of-sum-for-uncountable-indexing-set. – Martín-Blas Pérez Pinilla Nov 02 '19 at 21:00
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Thank you, I will read it more thoroughly later, but do we still call the objects $(a_j)_{i\in K}$ 'sequences' if $K$ uncountable, or something else to distinguish them from sequences as functions from $\mathbb{N}$? – Christopher.L Nov 02 '19 at 21:06
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@Christopher.L, "sequence" implies usually some order. – Martín-Blas Pérez Pinilla Nov 03 '19 at 10:24