Is the space of all bounded linear operators a subspace of the space of the linear operators?

Question

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 ^_^...

Answer 1

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.$