Application of Banach Separation theorem 
Let $(\mathcal{H},\langle\cdot,\cdot\rangle)$ be a Hilbert Space, $U\subset \mathcal{H},U\not=\mathcal{H}$ be a closed subspace and $x\in\mathcal{H}\setminus U$. Prove that there exists $\phi\in\mathcal{H}^*$, such that\begin{align}\text{Re } \phi(x)<\inf_{u\in U}\text{Re }\phi(u)
\end{align}
Hint: Observe that $\inf_{u\in U}\text{Re }\phi(u)\leq0$.

This seems like an application of the Banach Separation theorem. But the way I know it is not directly applicable. I know that for two disjoint convex sets $A$ and $B$ of which one is open there exists a functional seperating them. Is there anything special in this problem about $\mathcal{H}$ being Hilbert and not some general Banach space?

There is more general result which proof you can find in Rudin's Functional Analysis

Let $A$ and $B$ be disjoint convex subsets of topological vector space $X$. If $A$ is compact and $B$ is closed then there exist $\varphi\in X^*$ such that
  $$
\sup\limits_{x\in A}\mathrm{Re}(\varphi(x))<\inf\limits_{x\in B}\mathrm{Re}(\varphi(x))
$$

Your result follows if we take $X=\mathcal{H}$, $A=\{x\}$ and $B=U$.
Of course this is a sledgehammer for such a simple problem, because in case of Hilbert space we can explicitly say that functional
$$
\varphi(z)=\langle z, \mathrm{Pr}_U(x)-x\rangle
$$
will fit, where $\mathrm{Pr}_U(x)$ is the unique orthogonal projection of vector $x$ on closed subspace $U$.
