Multiple of Vector in Topological Vector Space Converges

Theorem

Let $K$ be a topological field.

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

Let $x \in X$.

Let $\lambda \in K$.

Let $\sequence {\lambda_n}_{n \mathop \in \N}$ be a sequence in $K$ such that $\lambda_n \to \lambda$.

Then:

$\lambda_n x \to \lambda x$

Proof

From the definition of a topological vector space, the map $f : K \times \struct {X, \tau} \to X$ defined by:

$\map f {\lambda, y} = \lambda y$

is continuous.

From Horizontal Section of Continuous Function is Continuous, we therefore have the map $c_x : X \to X$ defined by:

$\map {c_x} \lambda = \lambda x$

is continuous.

From Continuous Mapping is Sequentially Continuous, $c_x$ is sequentially continuous.

So we have:

$\lambda_n x \to \lambda x$

$\blacksquare$