Proof for definition of the direct sum of subspaces

Question

I don't get how this definition can describe direct sum of subspaces. What is (detailed/noob) explanation for this definition? What is proof for it?

For subspaces $$(P_1, . . . , P_k),$$ we define direct sum if $$\sum_{i=1}^{k} x_i \neq 0,$$ when $$x_i \in P_i$$ for $$i = {1 \ldots k}, $$ if there is $$ i_0 \in {1, . . . , k}, $$ that $$x_{i_0} \neq 0.$$

Answer 1

It means that the sum of the spaces is direct if the following property is satisfied:

For all $\;(x_1,\dots,x_k)\;$ such that $\;x_1\in P_1,\dots,\, x_k\in P_k$, if an $x_i$ is not $0$, then $\; x_1+\dots+x_k\ne0$.

Taking the contrapositive, it means that $\;\displaystyle\sum_{i=1}^k x_i$ is $0$ if and only if each $x_i$ is $0$.