Is is true that $||v+w||^2 = ||v||^2 + 2\langle v,w \rangle + ||w||^2$?

Question

Is it true that for a $\mathbb{R}$ vector space with dot product $\langle\cdot, \cdot\rangle$ and $||\cdot||$ norm

\begin{align} ||v+w||^2 = ||v||^2 + 2\langle v,w\rangle + ||w||^2 && \forall v,w \end{align}

and if so, why? Thank you!

Answer 1

Because in $\mathbb{R}^n$ (in particular when $n=1$), the norm comes from dot product: $||v||=\langle v,v \rangle ^{\frac{1}{2}}$, $\forall v\in \mathbb{R}$. Then, for $u,v\in \mathbb{R}$ we have: $$||u+v||^2=\langle u+v,u+v\rangle =\langle u,u+v\rangle +\langle v,u+v \rangle = $$ $$=\langle u,u\rangle +\langle u,v\rangle+\langle v,u\rangle+\langle v,v \rangle= $$ $$=||u||^2+2\cdot \langle u,v \rangle +||v||^2. $$

Answer 2

$||v+w||^2$

$=\langle v+w,v+w\rangle$

$=\langle v,v\rangle +\langle v,w\rangle +\langle w,v\rangle +\langle w,w\rangle$

$=\langle v,v\rangle +\langle v,w\rangle +\langle v,w\rangle +\langle w,w\rangle$

$=||v||^2+2\langle v,w\rangle+||w||^2$