Understanding Riesz representation theorem

Question

I was wondering about the vice-versa of the Riesz representation theorem. In the form that was presented to me, the theorem states that if $\phi(x): H\rightarrow\mathbb{C}$ is a continuous linear functional between a Hilbert space and the field of complex numbers, then we can find $x_0\in H$ such that $\phi(x)=(x_0,x)$ and $\|\phi\|=\|x_0\|$. Now, is it true that if we find such a vector, then the linear functional is continuous and thus limited? Thanks in advance!

Answer 1

Linearity is clear. To check continuity at zero, simply use the Cauchy-Schwarz inequality: if $x_n \to 0$ in $H$, then $|(x_0,x_n)| \leq \|x_0\| \|x_n\| \to 0$ because $\|x_n\| \to 0$.