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Suppose $V_1, \dots, V_m, W$ are vector spaces$^\dagger$. Prove that $\mathcal L(V_1 \times \cdots \times V_m,W)$ and $\mathcal L(V_1,W) \times \cdots \times \mathcal L(V_m,W)$ are isomorphic vector spaces.

I'm having some trouble with this one. If the vector spaces were finite-dimensional I could try proving that each has the same dimension, but the question doesn't specify that they're finite-dimensional spaces.

It seems like maybe I need to construct the isomorphism. So I'd need some linear mapping $$\big((v_1,\dots, v_m)\mapsto w\big)\mapsto (v_1 \mapsto w_1, \dots, v_m\mapsto w_m)$$ but TBH I'm getting a little confused from my own notation here.

Could someone give a hint as to how to get started on this exercise?

$\dagger$: The running assumption is that we're talking about $\Bbb R$- or $\Bbb C$-vector spaces, if that's relevant.

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    Notice that your attempt does not make sense, for $w_1$ does not mean anything. – 2017-02-13
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    Well, I didn't specify, but $w, w_1, \dots, w_m$ should all be elements of $W$ while $v_i$ should specifically be an element of $V_i$. – 2017-02-13
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    Yes. That does not make sense. – 2017-02-13

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Hints:

  • Do it for two factors first.

  • If $f:V_1\times V_2\to W$ is a linear map, then you can consider the map $f_1:v\in V_1\mapsto f(v,0)\in W$ and, similarly, a map $f_2:V_2\to W$ constructed from $f$. In this way you can produce a function $$\Phi:f\in L(V_1\times V_2,W)\mapsto (f_1,f_2)\in L(V_1,W)\times L(V_2,W).$$ Show that it does what you want.

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    Awesome-sauce! Thanks! – 2017-02-13