Start solutions to 3E

exercises/axler-3e.md

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+# Exercises from Axler 3.E
+
+3, 8, 13.
+
+## Question 3
+
+Suppose $V_1, \dots, V_m$ are vector spaces. Prove that
+$\mathcal{L}(V_1\times\dotsb\times V_m, W)$ and $\mathcal{L}(V_1, W)
+\times\dotsb\times \mathcal{L}(V_m, W)$ are isomorphic vector spaces.
+
+### Answer
+
+If all the vector spaces were finite-dimensional, we could show that
+the dimension of each side matched and be done. However, it is not
+stated that they are. So let us try to exhibit an isomorphism.
+
+An element of the left hand side, say $\rho$, is a map from tuples
+$(v_1, \dotsc, v_m)$ (where $v_i\in V_i$) to $W$. For each $i$, define
+$\rho_i\in \mathcal{L}(V_i, W)$ by $\rho_i(v_i) =
+\rho((0,\dotsc,0,v_i,0,\dotsc,0))$ for any $v_i\in V_i$. The tuple
+$(\rho_1,\dotsc,\rho_m)$ is an element of $\mathcal{L}(V_1, W)
+\times\dotsb\times \mathcal{L}(V_m, W)$,