Suppose $V_1,\dots,V_m$ are vector spaces such that $V_1\times\dots\times V_m$ is finite-dimensional. Prove that $V_j$ is finite-dimensional for each $j=1,\dots,m$.

Without loss of generality assume $j=1$. Consider the subspace $V_1 \times \{0\} \times \dots \times \{0\}$ of $V_1\times\dots\times V_m$.

This subspace is finite-dimensional (see 2.26), letting $T$ be the natural isomorphism from $V_1\times \{0\}\times\dots\times\{0\}$ to $V_1$ we see that $V_1$ is finite-dimensional as well.