(a) Let $(\mathcal{H},\langle\cdot, \cdot\rangle)$ be a real Hilbert space and let $B: \mathcal{H} \times \mathcal{H} \rightarrow \mathbb{R}$ be a bilinear map. If $B$ is continuous prove that there is an $M>0$ such that $|B(u, v)| \leqslant M\|u\|\|v\|$ for all $u, v \in \mathcal{H}$. [You may use any form of the Banach-Steinhaus theorem as long as you state it clearly.]

(b) Now suppose that $B$ defined as above is bilinear and continuous, and assume also that it is coercive: i.e. there is a $C>0$ such that $B(u, u) \geqslant C\|u\|^{2}$ for all $u \in \mathcal{H}$. Prove that for any $f \in \mathcal{H}$, there exists a unique $v_{f} \in \mathcal{H}$ such that $B\left(u, v_{f}\right)=\langle u, f\rangle$ for all $u \in \mathcal{H}$.

[Hint: show that there is a bounded invertible linear operator $L$ with bounded inverse so that $B(u, v)=\langle u, L v\rangle$ for all $u, v \in \mathcal{H}$. You may use any form of the Riesz representation theorem as long as you state it clearly.]

(c) Define the Sobolev space $H_{0}^{1}(\Omega)$, where $\Omega \subset \mathbb{R}^{d}$ is open and bounded.

(d) Suppose $f \in L^{2}(\Omega)$ and $A \in \mathbb{R}^{d}$ with $|A|_{2}<2$, where $|\cdot|_{2}$ is the Euclidean norm on $\mathbb{R}^{d}$. Consider the Dirichlet problem

$$-\Delta v+v+A \cdot \nabla v=f \quad \text { in } \Omega, \quad v=0 \quad \text { in } \partial \Omega$$

Using the result of part (b), prove there is a unique weak solution $v \in H_{0}^{1}(\Omega)$.

(e) Now assume that $\Omega$ is the open unit disk in $\mathbb{R}^{2}$ and $g$ is a smooth function on $\mathbb{S}^{1}$. Sketch how you would solve the following variant:

$$-\Delta v+v+A \cdot \nabla v=0 \quad \text { in } \Omega, \quad v=g \quad \text { in } \partial \Omega .$$

[Hint: Reduce to the result of part (d).]