Is this fact true $T:X \longrightarrow Y$ linear and bounded operator with $X$ reflexive then $T$ is compact?

Asked Apr 01 '15 at 18:57

Active Apr 01 '15 at 19:06

Viewed 97 times

I wonder if this fact is true.

I consider a linear operator $T:X \longrightarrow Y$, with $Y$ and $X$ two normed spaces. I suppose that $X$ is a reflexive space and that $T$ a bounded operator.

Is it true that $T$ is necessary a compact operator? If yes, how can I prove it?

Thank you!

edited Apr 01 '15 at 19:06

asked Apr 01 '15 at 18:57

GGG

1,765

3

Is every identity operator compact? – David Mitra Apr 01 '15 at 18:57
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No, if $X$ has not finite dimension....So the answer is no...thank you! – GGG Apr 01 '15 at 19:00
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You're welcome. By the way, the statement does hold for certain $X$ and $Y$. See this for example. – David Mitra Apr 01 '15 at 19:03

Is this fact true $T:X \longrightarrow Y$ linear and bounded operator with $X$ reflexive then $T$ is compact?

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