Kernel of Bounded Linear Transformation is Closed Linear Subspace

Theorem

Let $V, W$ be normed vector spaces.

Let $f: V \to W$ be a bounded linear transformation.

Then $\ker f$, the kernel of $f$, is a closed linear subspace of $V$.

Proof

By Kernel of Linear Transformation is Linear Subspace, $\ker f$ is a subspace of $V$.

By Continuity of Linear Transformations, $f$ is continuous.

Since $\ker f = f^{-1} \sqbrk{ \set{ \mathbf 0_W } }$, it follows from Continuity Defined from Closed Sets that $\ker f$ is closed.

Hence the result by definition of closed linear subspace.

$\blacksquare$