Translation of Local Basis in Topological Vector Space

Theorem

Let $K$ be a topological field.

Let $\struct {X, \tau}$ be a topological vector space over $K$.

Let $\sequence {U_\alpha}_{\alpha \mathop \in A}$ be a local basis at ${\mathbf 0}_X$.

Let $x \in X$.

Then $\sequence {U_\alpha + x}_{\alpha \in A}$ is an local basis at $x$.

Proof

From Translation of Open Set in Topological Vector Space is Open, $U_\alpha + x$ is an open neighborhood of $x$ in $\struct {X, \tau}$ for each $\alpha \in A$.

Let $U$ be an open neighborhood of $x$ in $\struct {X, \tau}$.

From Translation of Open Set in Topological Vector Space is Open, $U - x$ is an open neighborhood of ${\mathbf 0}_X$ in $\struct {X, \tau}$.

Since $\sequence {U_\alpha}_{\alpha \mathop \in A}$ is local basis at ${\mathbf 0}_X$, there exists $\beta \in A$ such that:

$U_\beta \subseteq U - x$

Then:

$U_\beta + x \subseteq U$

Since $U$ was arbitrary, we obtain that $\sequence {U_\alpha + x}_{\alpha \in A}$ is an local basis at ${\mathbf 0}_X$.

$\blacksquare$