Given a normed vector space $X$ such that the norm can be derived from an inner product ( pre Hilbert space ) and $S\subset X$ a subspace of $X$, then generally $\overline{S}\neq S ^{\bot\bot}.$ Also in general, $\overline{S}\oplus S ^{\bot}\neq X. $
I am wondering if the case $S=\{(x,y)|x, y \in R^2, x>0\}\subset R^2$ is a good example of the first situation ? Can you give some other examples ? How about examples for the second situation ?
Many thanks in advance.
As an example of the second situation, take $X$ as the set of all sequences $(a_n)_{n\in\mathbb N}$ of real numbers such that $n\gg1\implies a_n=0$. Consider the inner product$$\bigl\langle(a_n)_{n\in\mathbb N},(b_n)_{n\in\mathbb N}\bigr\rangle=\sum_{n=1}^\infty a_nb_n.$$Let$$S=\left\{(a_n)_{n\in\mathbb N}\in X\,\middle|\,\sum_{n=1}^\infty\frac{a_n}{n^2}=0\right\}.$$and note that $S\neq X$. Then $S^\perp=\{0\}$ and therefore $S\oplus S^\perp\neq X$.