$ \begin{array}{l} \text{17. Suppose that } V_1, \ldots, V_m \text{ are finite-dimensional subspaces of } V. \text{ Prove that} \\ V_1 + \ldots + V_m \text{ is finite-dimensional and} \\ \text{dim} (V_1 + \ldots + V_m) \leq \text{dim} V_1 + \ldots + \text{dim} V_m . \\ \text{The inequality above is an equality if and only if } V_1 + \ldots + V_m \text{ is a direct} \\ \text{sum, as will be shown in 3.94.} \\\\ \text{SOLUTION: For each } k = 1, \ldots, m, \text{ choose a basis of } V_k . \\\text{ Put these bases together} \text{to form a single list of vectors in } V.\\ \text{ Clearly this list spans } V_1 + \ldots + V_m , \text{ and thus } V_1 + \ldots + V_m \text{ is finite-dimensional.}\\\text{ Furthermore, the dimension of } V_1 + \ldots + V_m \text{ is less than or equal to the number of vectors} \\\text{ in this list (by 2.30), which equals} \\\\ \text{dim} V_1 + \ldots + \text{dim} V_m .\\\\ \text{ In other words,} \\ \text{dim} (V_1 + \ldots + V_m) \leq \text{dim} V_1 + \ldots + \text{dim} V_m . \\ \text{Edward Frenkel} \end{array} $