Category:Direct Product of Topological Vector Spaces is Hausdorff iff Hausdorff Factor Spaces

This category contains pages concerning Direct Product of Topological Vector Spaces is Hausdorff iff Hausdorff Factor Spaces:

Let $K$ be a topological field.

Let $I$ be a set.

Let $\family {X_i}_{i \in I}$ be an $I$-indexed family of topological vector spaces over $K$.

Let:

$\ds X = \prod_{i \mathop \in I} X_i$

be the direct product of $\family {X_i}_{i \in I}$.

Equip $X$ with the product topology.

Then $X$ is Hausdorff if and only if $X_i$ is Hausdorff for each $i \in I$.