I am looking for a specific name for the topology induced by metric on $\ell^2$. This space has its "standard" topology generated by $\sqrt{\sum_{i=1}^{\infty}(x_i-y_i)^2}$. With such topology, the space can be called Hilbert space.
What do we usually call this topology? And is it similar topology as on $\ell^p$ spaces?
Also, the vector space $\mathbb{R^{\infty}}$ is endowed with the Tychonoff product topology and $\ell^2$ should be a subset of this space. Does it mean that $\ell^2$ "inherits" the Tychonoff product topology or it is at least possible to give to $\ell^2$, even though it is not the standard choice?