Chapter 5 - Inner Product Spaces - 5.1 Definition of an Inner Product Space - Problems - Page 351: 30

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Assume $V$ be an inner product space. Since $u,v,w \in V$ we have $$(u,v+w)=\bar{(v+w,u)}=\bar {(v,u)+(w,u)}=\bar {(v,u)+(w,u)}=(u,v)+(u,w) $$ Hence, in any inner product space $V, ⟨v, 0⟩ = 0$ for all $v$ in $V$.