Dilation of Closed Set in Topological Vector Space is Closed Set/Proof 2

Proof
Define a mapping $c_\lambda : X \to X$ by:
 * $\map {c_\lambda} x = \lambda x$

for each $x \in X$.

From Dilation Mapping on Topological Vector Space is Homeomorphism, $c_\lambda$ is a homeomorphism.

From Definition 4 of a homeomorphism, $c_\lambda$ is therefore a closed mapping.

Hence $c_\lambda \sqbrk F = \lambda F$ is closed.