Question: For every Banach space $X$ and its subspace $Y$, is there a complemented subspace $Z$ in $X$ such that $Y \subset Z \subset X $ and $\operatorname{card}(Y)=\operatorname{card}(Z)$ i.e., $Y$ and $Z$ have the same cardinality?
The answer is yes if $\operatorname{card}(X)=\mathfrak c$ (i.e., Continuum), the proof is easy by taking a hyperplane $H$ such that $Y \subset H \subset X$, then $\operatorname{card}(Y)=\operatorname{card}(H)=\mathfrak c$.
But I don't know whether it holds for general spaces.