Vector Subspace of Topological Vector Space is Topological Vector Space

Theorem

Let $K$ be a topological field.

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

Proof

From the definition of a vector subspace, $\struct {Y, +_Y, \circ_Y}_K$ is a vector space over $K$.

Since $\struct {\struct {X, +_X, \circ_X}_K, \tau_X}$ is a topological vector space, $+_X : X \times X \to X$ and $\circ_X : K \times X \to X$ are continuous.

From Restriction of Continuous Mapping is Continuous, $+_Y : Y \times Y \to Y$ and $\circ_Y : K \times Y \to Y$ are continuous.

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

$\blacksquare$