Dilation Mapping on Topological Vector Space is Homeomorphism

Theorem

Let $K$ be a topological field.

Let $X$ be a topological vector space over $K$.

Let $\lambda \in K \setminus \set {0_K}$.

Let $c_\lambda$ be the dilation by $\lambda$ mapping.

Then $c_\lambda$ is a homeomorphism.

Proof

From Dilation Mapping on Topological Vector Space is Continuous, both $c_{\lambda}$ and $c_{1/\lambda}$ are continuous.

It is therefore sufficient to establish that $c_{1/\lambda}$ is the inverse mapping of $c_\lambda$.

For all $x \in X$, we have:

\(\ds \map {\paren {c_\lambda \circ c_{1/\lambda} } } x\)

\(=\)

\(\ds \lambda \paren {\frac 1 \lambda x}\)

\(\ds \)

\(=\)

\(\ds x\)

and:

\(\ds \map {\paren {c_{1/\lambda} \circ c_\lambda} } x\)

\(=\)

\(\ds \frac 1 \lambda \paren {\lambda x}\)

\(\ds \)

\(=\)

\(\ds x\)

So both $c_\lambda \circ c_{1/\lambda}$ and $c_{1/\lambda} \circ c_\lambda$ are the identity mapping for $X$.

So $c_{1/\lambda}$ is the inverse mapping of $c_\lambda$, as required.

$\blacksquare$

Sources

• 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $1.6$: Topological vector spaces