= Solution
On $\widetilde H_m$, use the inner product
$$
(v,w)_{\widetilde V}=(\widetilde Av,w)_{\widetilde H}.
$$
The map in the question is continuous, and part a gives
$$
\begin{aligned}
(\Phi_m(v),v)_{\widetilde V}
&=-\|v\|^2
-\frac1\nu\langle\widetilde B(v,v),v\rangle
+\frac1\nu(f,v)\\
&=-\|v\|^2+\frac1\nu(f,v).
\end{aligned}
$$
On the sphere $\|v\|=\rho$ with any $\rho>R$,
$$
(\Phi_m(v),v)_{\widetilde V}
\leq-\rho^2+\frac{|f|}{\nu\mu_1^{1/2}}\rho<0.
$$
The <Brouwer inward-pointing zero lemma> therefore supplies $v^*$ in the ball with $\Phi_m(v^*)=0$. Multiplication by $\nu\widetilde A$ shows that this zero satisfies
$$
\nu\widetilde Av^*
+\widetilde P_m\widetilde B(v^*,v^*)
=\widetilde P_mf.
$$
Thus every Galerkin system has at least one solution.
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