Canonical Isomorphism Between m-Tuple Vector Space and Vector Space This is a question from Loomis and Sternberg's Advanced Calculus. It is Chapter 1 question 3.38.
The question is:
Given an $m$-tuple of vector spaces $\{W_i\}_{i=1}^m,$ suppose that there are vector space $X$ and maps $p_i$ in $Hom(X,W_i),$ $i=1,...,m$ with the following property:
Property: For any $m$-tuple of linear maps $\{T_i\}$ with common domain space $V$ to the above spaces $W_i$ (so $T_i\in Hom(V,W_i$)), there is a unique $T$ in $Hom(V,X)$ such that $T_i=p_i\circ T, i=1,...m.$
Prove that there is a canonical isomorphism from
$$W=\Pi_{1}^m W_i \quad\text{ to } \quad X$$
under which the given map $p_i$ becomes the projections $\pi_i.$
Now, the obvious mapping would be to map $x\in X$ to the $m$-tuple $(p_1(x), p_2(x),...p_m(x))$ in $W$. However, I'm struggling to show that this is an isomorphism. I would like to find an inverse of this map from $W$ to $X$, but I'm struggling to do so. I think I'm missing something that allows me to invert $T_i.$
I'd prefer not to have this solved, but a nudge in the right direction would be appreciated.
 A: An over-the-top nudge might be for me to suggest you dig into a little category theory. With the general principle of 'universal properties make objects unique up to isomorphism', your question becomes clear. Should you be curious, read through the introduction of this, and go on to read the whole book if you like the sound of it...
Anyway, a concrete hint:

Your obvious mapping is correct. In introductory category theory generally, the obvious thing is almost always the correct thing.
You want to show your map is an isomorphism, let's call it $\Gamma:W\to\prod_{k=1}^mX$. That means two things:

*

*There exists a map $\Lambda:\prod_{k=1}^mX\to W$

*$\Lambda\circ\Gamma=\mathrm{id}_W$ and $\Gamma\circ\Lambda=\mathrm{id}_{\prod X}$.

Verify the property you're given for $W$ also applies to $\prod_{k=1}^mX$, if you haven't already. Use this property, for $W$ and for $\prod_{k=1}^mX$, and use nothing else (!) to a) obtain $\Lambda$ and b) show they are mutually inverse. Think about the way that we have a correspondence between product maps and their components - do the components completely determine the product map? Is that useful?

Note that you know nothing a priori about $W$ other than this property. Do not try to appeal to elements of $W$ or try any fancy linear algebra - you don't need it. Really, do use nothing else but this property.
