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Suppose we have a vector space $W$ and $V_i,i=1,...,k$ are subspaces of $W$. When proving that $W=V_1+V_2+...+V_k$ we show that $\forall x \in W$ holds that $x=μ_1υ_1+μ_2υ_2+...+μ_kυ_k$, where $ υ_i \in V_i$ for some $μ_ι \in \Bbb F$ or we show that $dim(W)=$$\sum_{i=1}^kdim(V_i)$ if we already know that $V_i \cap(V_1+V_2+...+V_{i-1}+V_{i+1}+...+V_k)=\{0\}, \forall i \in \{1,...,k\}$

Are there more ways to prove the sum?

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    This list covers the idea behind most ways to check equality of vector spaces. However, one can reinterpret this list in many ways and come up with techniques that appear different, even though they are, essentially, doing the same types of tests. – 2017-01-02
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    This makes sense. Many thanks. – 2017-01-02

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