# Finite Rank Operators Dense in Compact Linear Transformations

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## Theorem

Let $H, K$ be Hilbert spaces.

Then:

- $\map {B_{00} } {H, K}$ is everywhere dense in $\map {B_0} {H, K}$

where:

- $\map {B_{00} } {H, K}$ is the space of continuous finite rank operators from $H$ to $K$
- $\map {B_0} {H, K}$ is the space of compact linear transformations from $H$ to $K$.

That is, for every $T \in \map {B_0} {H, K}$, there is a sequence $\sequence {T_n}_{n \mathop \in \N}$ in $\map {B_{00} } {H, K}$ such that:

- $\ds \lim_{n \mathop \to \infty} \norm {T_n - T} = 0$

where $\norm {\, \cdot \,}$ denotes the norm on bounded linear transformations.

## Proof

This theorem requires a proof.In particular: Proceeds by sequence definition of limit pointYou 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) $\text {II}.4.4$