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.