Inner product space expansions

Question

I would like to expand the following two:

i). $||v+w||^2$

ii). $||v-w||^2$

I know that $||v||=\sqrt{\langle v,v \rangle}$.

For (i) I would therefore have

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

and for (ii) I would have

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

Is this correct?

Answer 1

hint

Norms and inner product are related by $||a||^2=\langle a, a\rangle$. So \begin{align*} ||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 \end{align*}