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quasar987

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## Main Question or Discussion Point

I have a theorem here that I find a little surprising and I would like confirmation that I am interpreting it correctly.

The theorem says that for E an

In particular, this means that as soon as E if infinite dimensional and T:E-->E is compact, then T is not invertible. There are

The theorem says that for E an

**infinite dimensional**Banach space (over K=R or C) and T:E-->E a compact operator, 0 is in the spectrum of T. That is to say, [tex]0\in\sigma(T)=\{\lambda\in K:\lambda I-T \mbox{ is not invertible}\}[/tex].In particular, this means that as soon as E if infinite dimensional and T:E-->E is compact, then T is not invertible. There are

*no*invertible compact operators on infinite dimensional Banach spaces!