Inner product in Hilbert spaces

Question

Considering a sequence $\{\boldsymbol{v}_k\}_{k=1}^\infty$ in a Hilbert space $\mathcal{H}$, and let $\{c_k\}_{k=1}^\infty \in \ell^2(\mathbb{N})$.

Then for all $\boldsymbol{v}\in\mathcal{H}$

$$ \langle \sum_{k=1}^\infty c_k \boldsymbol{v}_k, \boldsymbol{v}\rangle_\mathcal{H} = \sum_{k=1}^\infty c_k \langle \boldsymbol{v}_k, \boldsymbol{v} \rangle _\mathcal{H} $$

Anyone who can assist me understanding this equal sign, or provide some intermediate results to prove this ?

Thanks in advance.

Answer 1

The more direct thing to say is simply that if $w_k\to w$ in $\cal H$ then $\langle w_k,v\rangle\to\langle w,v\rangle$ in $\Bbb C$ for any chosen vector $v\in \cal H$. Then we can simply apply this to the sequence of partial sums.

To show $\|w_k-w\|\to0$ implies $|\langle w_k,v\rangle-\langle w,v\rangle|\to0$, notice that

$$|\langle w_k,v\rangle-\langle w,v\rangle|=|\langle w_k-w,v\rangle|\le \|w_k-w\|\cdot \|v\| $$

by the Cauchy-Schwarz identity.