Operator norm estimate 
Let $H$ be a Hilbert space with orthonormal basis  $(e_{j})_{j\in\mathbb{N}}$. Furthermore, let $B\colon H\rightarrow C[a,b]$ be a bounded operator. According to the Riesz-Frechet theorem there is a unique $k_{x}\in H$ with $\langle h,k_x\rangle=(Bh)(x)$ and all $h \in H$.
Show that $\|k_{x}\|\leq \|B\|_{H\rightarrow C[a,b]}<\infty.$

I am really confused by this if I use Cauchy-Schwarz inequality I don't get anywhere
$$ \|Bh\|=\langle h,k_{x}\rangle|\leq\|h\|\|k_{x}\|.$$
or
$$\|Bh\|\leq\|B\|\|h\|.$$
How do I proceed? I don't see how I would get the inequality or that it is $<\infty$?

While this doesn't apply to all problems involving inner product spaces, one of the common ways to estimate the norm of something is to instead estimate its norm squared:
$$ \|k_x\|^2 = | \langle k_x, k_x \rangle| = |Bk_x(x)| \le \|Bk_x\|_{C[a,b]} \le \|B\|_{op} \|k_x\|$$
where the operator norm $\|B\|_{op}$ is finite by assumption.
