The Norm of a Real Inner Product Space

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $V$ be a real inner product space with inner product $\langle\cdot,\cdot\rangle$, induced norm $\|\cdot\|$, zero vector $0_V$ and differences $u-v$. Let $u,v,w\in V$ and $\lambda\in\mathbb{R}$.

(i) Bilinearity and the zero vector.

$$\langle u,v+w\rangle=\langle u,v\rangle+\langle u,w\rangle,\qquad \langle u,\lambda v\rangle=\lambda\,\langle u,v\rangle,\qquad \langle 0_V,v\rangle=\langle v,0_V\rangle=0.$$

(ii) Expansion of squares.

$$\|u+v\|^{2}=\|u\|^{2}+2\,\langle u,v\rangle+\|v\|^{2},\qquad \|u-v\|^{2}=\|u\|^{2}-2\,\langle u,v\rangle+\|v\|^{2}.$$

(iii) Normed space. $\|\cdot\|$ is a norm on $V$, so $V$ with $\|\cdot\|$ is a real normed space.

(iv) Parallelogram law.

$$\|u+v\|^{2}+\|u-v\|^{2}=2\,\|u\|^{2}+2\,\|v\|^{2}.$$

(v) Continuity of the inner product. Let $d_V$ be the norm metric of the real normed space of (iii), a metric on $V$ by The Norm Metric of a Real Normed Space (ii). If $(u_m)$ and $(v_m)$ are sequences in $V$ that converge to $u$ and $v$ in $(V,d_V)$, then $\langle u_m,v_m\rangle\to\langle u,v\rangle$ in $\mathbb{R}$, in the sense of Convergent Sequences of Real Numbers (i).