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
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Sources
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\text {II}.4.4$