Let $\mathcal{H}$ be a Hilbert space, and $\mathcal{H}^*$ its topological dual space (the space of continuous linear forms on $\mathcal{H}$). The exists a conjugate-linear isometry between these two Hilbert spaces, so it stands to reason that
$$\langle f_\vec{v}|f_\vec{w}\rangle \space = \langle\vec{w}|\vec{v}\rangle$$
would be the inner product on $\mathcal{H}^*$, and thus
$$\lVert f_\vec{v}\rVert = \sqrt{\langle\vec{v}|\vec{v}\rangle} = \lVert\vec{v}\rVert.$$
But I've been taught that the norm of a linear form on a Hilbert space is
$$\lVert f_\vec{v}\rVert = \mathrm{sup}\left\{\frac{|f_\vec{v}(\vec{x})|}{\lVert\vec{x}\rVert}\space|\space \vec{x}\in\mathcal{H}\smallsetminus\{0\}\right\}.$$
Are these two norms equivalent? How could I prove it?