Space of Compact Linear Transformations is Banach Space

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Theorem

Let $H, K$ be Hilbert spaces.

Let $\map {B_0} {H, K}$ be the space of compact linear transformations from $H$ to $K$.

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

Now $\map {B_0} {H, K} \subseteq K^H$, the set of mappings from $H$ to $K$.

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

Let $\norm {\, \cdot \,}$ denote the norm on bounded linear operators.

Then $\norm {\, \cdot \,}$ is a norm on $\map {B_0} {H, K}$.

Furthermore, $\map {B_0} {H, K}$ is a Banach space with respect to this norm.

Proof

This theorem requires a proof. In particular: Subspace of $\map B {H, K}$; 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) $\text {II.4.2 (b)}$