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 $\mathbb {R} ^{\infty }$ or $\mathbb {R} ^{\omega }$. 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 $\mathbb {C} ^{\omega }$.