Vector Subspace of Hausdorff Topological Vector Space is Hausdorff Topological Vector Space

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Theorem

Let $K$ be a topological field.

Let $\struct {\struct {X, +_X, \circ_X}_K, \tau_X}$ be a Hausdorff topological vector space over $K$.

Let $Y$ be a vector subspace of $X$.

Let $\tau_Y$ be the subspace topology on $Y$ induced by $\tau_X$.

Let $+_Y : Y \times Y \to Y$ be the restriction of $+_X$ to $Y \times Y$.

Let $\circ_Y : K \times Y \to Y$ be the restriction of $\circ_X$ to $K \times Y$.


Then $\struct {\struct {Y, +_Y, \circ_Y}_K, \tau_Y}$ is a Hausdorff topological vector space.


Proof

From Vector Subspace of Topological Vector Space is Topological Vector Space, $\struct {\struct {Y, +_Y, \circ_Y}_K, \tau_Y}$ is a topological vector space.

Since $\struct {\struct {X, +_X, \circ_X}_K, \tau_X}$ is a Hausdorff topological vector space, it is in particular Hausdorff.

By Subspace of Hausdorff Space is Hausdorff, since $\struct {\struct {Y, +_Y, \circ_Y}_K, \tau_Y}$ is a topological subspace of $\struct {\struct {X, +_X, \circ_X}_K, \tau_X}$, we have that $\struct {\struct {Y, +_Y, \circ_Y}_K, \tau_Y}$ is Hausdorff.

So $\struct {\struct {Y, +_Y, \circ_Y}_K, \tau_Y}$ is a Hausdorff topological vector space.

$\blacksquare$

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