Given $W$ a vector space and $V$ a proper subspace of $W$, which has a basis set with cardinality uncountable infinity.
Can the function $F:\mathcal W \to \mathcal V$, ever be bijective, for some pair of $(\mathcal W, \mathcal V)$? $$$$Where $\mathcal W, \mathcal V$ denotes an arbitrary basis of space $W$ and $V$ respectively?
If instead of uncountable basis set we have atmost countable basis then the answer is a clearly NO.
But how to argue rigorously for uncountable case?
How to characterise the set of proper subspaces $V$ such that such a bijection $F$ exists, for each $W$?
Thanks.