Adjoint of Finite Rank Operator

Theorem

Let $H, K$ be Hilbert spaces.

Let $T \in \map {B_{00} } {H, K}$ be a bounded finite rank operator.

Then:

$T^* \in \map {B_{00} } {K, H}$

that is, the adjoint of $T$ is also a bounded finite rank operator.

Proof

This theorem requires a proof. 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) $\text{II}.4$ Exercise $3$