Sorry for kind of clickbait title, but I am very interested in $l^2$ space - space of all square-summable (and ths convergent) sequences.
First, since I have read that $l^2$ is (with a pinch of salt) something like generalized Euclidean space or "sum of real lines" (really large sum though) - how can we really relate $l^2$ spaces and Euclidean spaces, particularly $\mathbb{R}$ or $\mathbb{R^n}$? Can we really say statements like that?
Second, how does the "standard" topology on $l^2$ look like? Can we deliver it somehow from Euclidean standard topology?
Third, how are Hilbert spaces related to Euclidean generally? (Since $l^2(E)$ is isometric to any other Hilbert spaces, as I already asked about here).
I hope these questions are not too ambiguous - I am always interested in intuition more than the technical stuff). I can try to make them more precise.
Thank you for your insights.
The linear $\ell^2$ has infinite sequences as vectors, namely precisely those that are square summable so that $\sum_{i=0}^\infty x_i^2$ converges and this value being finite allows us to define a norm $\|x\|_2 = \sqrt{\sum_{i=0}^\infty x_i^2}$ which has the usual properties of a norm, like we know it from the Euclidean norm on $\Bbb R^n$. We can embed $\Bbb R^n$ into $\ell^2$ by $(x_1, \ldots, x_n) \to (x_1, \ldots x_n, 0,0,0,\ldots)$ and we have an isometry, so the finite Euclidean spaces sit nicely inside this $\ell^2$. It also has an inner product (so we can talk about orthogonal vectors and such geometric concepts; we can define angles between vectors as usual etc.). The topology is also induced by the metric from this norm as $d(x,y)= \|x-y\|_2$ etc. It's also separable and complete like $\Bbb R^n$ is and locally convex, path-connected and all such nice things.
But in infinite dimensional spaces things do behave differently: no compact set can have a non-empty interior (so no local compactness) and if $K$ is compact (or even $\sigma$-compact) it can be shown that $\ell^2 \simeq \ell^2\setminus K$, in stark contrast with $\Bbb R^n$ where even removing a point leaves us with a non-homeomorphic result... The book about infinite-dimensional topology by Bessaga and Pelczynski is a classic intro to such facts.