First Isomorphism Theorem/Topological Vector Spaces

Theorem

Let $K$ be a topological field.

Let $\struct {X, \tau_X}$ and $\struct {Y, \tau_Y}$ be vector spaces over $K$.

Let $T : X \to Y$ be a continuous and open linear transformation.

Let $\ker T$ be the kernel of $T$.

Let $X/\ker T$ be the quotient topological vector space of $X$ modulo $\ker T$.

Then $X/\ker T$ is topologically isomorphic to $\Img T$.

Proof

Let $\pi : X \to X/\ker T$ be the quotient mapping.

From First Isomorphism Theorem: Vector Spaces, there exists a linear isomorphism $\Lambda : X/\ker T \to \Img T$ with $T = \Lambda \circ \pi$.

It remains to argue that $\Lambda$ is a homeomorphism.

Since $T$ is continuous, we obtain that $\Lambda$ is continuous from Factorization of Continuous Linear Transformation between Topological Vector Spaces.

Since $T$ is open, we obtain that $\Lambda$ is open from Factorization of Open Linear Transformation between Topological Vector Spaces.

From Bijection is Open iff Inverse is Continuous, $\Lambda^{-1}$ is continuous.

So $\Lambda$ is a homeomorphism, and hence a topological isomorphism.

$\blacksquare$