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Maybe I'm mistaken but I think that's the question that our professor asked us today: Let E and F be two normed vector spaces, prove that the space of all bounded linear operators from E to F is a subspace of the space of the linear operators from E to F.

is it a legit question and if so how can I proceed?

Thank you ^_^...

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  • $\begingroup$ Why do you doubt that the question is legitimate? What do you know about proving a subset of a vector space is a subspace? $\endgroup$ – Ethan Bolker Nov 20 '16 at 18:29
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You need to show that:

  • $0$ is bounded,
  • if $f$ and $g$ are bounded then $f+g$ is bounded, and
  • if $f$ is bounded and $c$ is a scalar then $cf$ is bounded.

Note that $\|(f+g)(v)\|_F\le \|f(v)\|_F+\|g(v)\|_F, \forall v\in E$ and $ \|cf(v)\|_F=|c| \|f(v)\|_F, \forall v\in E.$

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  • $\begingroup$ Thank you very much I think I can see the bottom of it now... $\endgroup$ – Cynic Yahya Nov 20 '16 at 18:35

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