Space of Compact Linear Transformations is Banach Space

Theorem

Let $H, K$ be Hilbert spaces, and let $B_0 \left({H, K}\right)$ be the space of compact linear transformations from $H$ to $K$.

Let $\Bbb F \in \left\{{\R, \C}\right\}$ be the ground field of $K$.

Now $B_0 \left({H, K}\right) \subseteq K^H$, the set of mappings from $H$ to $K$.

Therefore, $B_0 \left({H, K}\right)$ can be endowed with pointwise addition ($+$) and ($\Bbb F$)-scalar multiplication ($\circ$).

Let $\left\Vert{\cdot}\right\Vert$ denote the norm on bounded linear operators.

Then $\left\Vert{\cdot}\right\Vert$ is a norm on $B_0 \left({H, K}\right)$.

Furthermore, $B_0 \left({H, K}\right)$ is a Banach space with respect to this norm.

Proof

This theorem requires a proof. In particular: Subspace of $B(H,K)$; completeness is nontrivial. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $II.4.2(b)$