Let $(e_i)_{i\in\Bbb N}$ be an orthonormal basis for Hilbert space $\mathcal{H}$. Consider $$\mathcal{K}_1:=\overline{\operatorname{span}}(e_1+ e_2, e_2, e_3, e_5, e_7,\ldots)$$ and $$\mathcal{K}_2:=\overline{\operatorname{span}} \left( \sum_{i=1}^\infty \frac{1}{2^{i+1}} e_{2i}, e_1+ e_2, e_3+ e_4, e_5+ e_6,\ldots \right).$$ I want to find any relation between this two subspaces. Is one is a subspace of another?
For this purpose, I put $e_1 = \sum c_i(e_{2i-1} + e_{2i}) + \sum \frac{d}{2^{i+1}} e_{2i}$. But it gives $c_1=0 , c_1= 1$.
Hint: The first space is nothing but the closed subspace spanned by $e_1,e_2,e_3,e_5,e_7...$. So its orthogonal complement is the closed subspace spanned by $e_4,e_6,...$. Show that the orthogonal complement of the second space consist if $(x_1,-x_1,x_3,-x_3,x_5,-x_5,...)$ with $\sum x_i2^{-i-1} =0$. It is fairly easy to see that neither of the orthogonal complements is contained in the other. Hence there is no inclusion between the given subspaces.