Hilbert Sequence Space is Homeomorphic to Countable Infinite Product of Real Number Spaces
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Theorem
Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\ds \sum_{i \mathop \ge 0} x_i^2$ 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 $\R^\omega$.
Proof
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Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $36$. Hilbert Space: $6$