Suppose that $T$ is a linear operator from a normed linear space $ X $ into normed linear space $ Y $.Prove that the following are equivalent.

$1)$ The operator $T$ is continuous.

$2)$The set $ T(K)$ is a weakly compact subset of $Y$ whenever $K$ is a weakly compact subset of $X$.

1) to 2) is clear but 2) to 1)??

  • $\begingroup$ The answer here may help. $\endgroup$ – David Mitra Feb 26 '15 at 9:43
  • $\begingroup$ You could add iii: $T(K)$ is bounded for every compact subset of $X$. Since weakly compact sets are bounded, ii implies iii which gives i because you only have to prove that images of null sequences are bounded. $\endgroup$ – Jochen Feb 26 '15 at 10:02

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