Sum of Closed Linear Subspace and Finite-Dimensional Subspace of Hausdorff Topological Vector Space

Theorem

Let $\GF \in \set {\R, \C}$.

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

Let $N$ be a closed linear subspace of $X$.

Let $F$ be a finite dimensional linear subspace of $X$.

Then $N + F$ is closed in $\struct {X, \tau}$.

Proof

Let $\struct {X/N, \tau_N}$ be the quotient topological vector space of $X$ modulo $N$.

From Characterization of Hausdorff Topological Vector Space, $X/N$ is Hausdorff.

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

From Image of Linear Transformation is Submodule and Dimension of Image of Vector Space under Linear Transformation is Bounded Above by Dimension of Vector Space, we have that $\pi \sqbrk F$ is a finite dimensional linear subspace of $\struct {X/N, \tau_N}$.

From Finite-Dimensional Subspace of Hausdorff Topological Vector Space is Closed, $\pi \sqbrk F$ is closed in $\struct {X/N, \tau_N}$.

From the definition of the quotient topology, $\pi$ is continuous.

So $\pi^{-1} \sqbrk F$ is closed in $\struct {X/N, \tau_N}$.

From Preimage of Image of Linear Transformation, we have:

$\pi^{-1} \sqbrk {\pi \sqbrk F} = \ker \pi + F$

From Kernel of Quotient Mapping, we have $\ker \pi = N$ and so:

$\pi^{-1} \sqbrk {\pi \sqbrk F} = N + F$

Hence $N + F$ is closed in $\struct {X/N, \tau_N}$.

$\blacksquare$

Sources

• 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $1.41$: Theorem