Countable-dimensional real vector space

Definition
We define countable-dimensional real vector space as a vector space over the field of real numbers with countable dimension. This space is denoted $$\R^\infty$$ or $$\R^\omega$$. It is equipped with the product topology from the usual Euclidean topology on the real line.

Note that, as a topological space, this is homeomorphic to countable-dimensional complex vector space $$\mathbb{C}^\infty$$ or $$\mathbb{C}^\omega$$.