Hilbert Sequence Space is Homeomorphic to Countable Infinite Product of Real Number Spaces

Theorem
Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\displaystyle \sum_{i \mathop \ge 0} x_i^2$ is convergent.

Let $\ell^2 = \struct {A, d_2}$ be the Hilbert sequence space on $\R$.

Let $\struct {\R, \tau_d}$ denote the real number line under the Euclidean topology.

Let $\R^\omega = \ds \prod_{i \mathop \in \N} \struct {\R, \tau_d}$ denote the countable-dimensional real Cartesian space under the product topology.

Then $\ell^2$ is homeomorphic to $\R^\omega$.