I'm not sure how to show that the set of all bounded multilinear maps is a vector space. Could someone help me?

  • $\begingroup$ Is the sum of two bounded multilinear maps a bounded multilinear map? If you multiply by a scalar, does it change whether something is bounded or multilinear? Is there a zero object? There's a list of requirements for something to be a vector space - have you tried checking each one? $\endgroup$ – user296602 Jan 24 '16 at 18:24

Hint: First, you want to show that the $0$ map is bounded and multilinear.

Then, you need to demonstrate that if $f,g$ are bounded multilinear maps, and $\alpha$ is an element of your original field, then $\alpha f + g$ is also a bounded multilinear map.

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