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Let W1,W2,....Wk be subspaces of vector space V over field F. Prove that the span of the union of those subspaces is equal to the sum of those subspaces. I know we can prove this by showing that the span of the union of the subspaces is a subset of the sum of the subspaces and vice versa. However, I don't know how to prove the span of the union is a subset of the sum. How do I do that?

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    Isn't this pretty much a direct application of the definition? (definition of sum of vector spaces.) – 2017-02-09

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Let $W=W_1 \cup ....\cup W_k$ and $x \in span(W)$. Then there are $x_1,...,x_m \in W$ and $a_1,...,a_m \in F$ such that

$x=a_1x_1+...+a_mx_m$.

Since $x_j \in W$, there is $n_j \in \{1,2,...,k\}$ with $x_j \in W_{n_j}$. Hence $a_j x_j \in W_{n_j}$ and therefore

$$x \in W_{n_1}+W_{n_2}+...+W_{n_m} \subseteq W_1+...+W_k.$$