Find an example for a prehilbert space $X$ and a subspace $U$ with $\overline{U} \neq U^{\perp \perp}$ and $\overline{U} \bigoplus U^\perp \neq X$.
The prehilbertspace does not have to fullfil both conditions, it's ok to find one for each condition. Would appreciate any help!
Let $X$ consist of sequences $(a_1,a_2,\cdots)\in \ell^2(\mathbb{N})$ with only a finite number of non-zero entries. This is a pre-Hilbert space. Let $U = \{ f = (f_n) \in X : \sum_{n=1}^{\infty}\frac{1}{n}f_n = 0 \}$.
There is no non-zero element $f$ of $X$ in $U^{\perp}$ because any such $f$ must satisfy $$ 0=\langle f, (1,-2,0,0,\cdots)\rangle = f_1-2f_2 \\ 0=\langle f, (1,0,-3,0,\cdots)\rangle = f_1-3f_3 \\ ... $$ leading to $f=f_1(1,\frac{1}{2},\frac{1}{3},\cdots)$, which is not in $X$ unless $f_1=0$ and, hence, $f=0$. So $U^{\perp}=\{0\}$ and $U^{\perp\perp}=X$. However, $\overline{U}\ne X$ because $(1,0,0,0,\cdots) \notin \overline{U}$. So $X=U^{\perp\perp}\ne \overline{U}$. And $\overline{U}\oplus U^{\perp} \ne X$.