3
votes
1answer
53 views

Banach space with cardinality bigger than $\mathfrak{c}$.

By using the infromation contained in this post, we have that the cardinality of every Banach space is equal to its dimension, which in turn, is bigger or equal to $\mathfrak{c}$. In my area of ...
5
votes
1answer
127 views

One problem about complemented subspace

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., $Y$ ...
15
votes
1answer
736 views

Cardinality of a Hamel basis

What is the cardinality of a Hamel basis of $\ell_1(\mathbb{R})$? Is it deducible in ZFC that it is seemingly continuum? Does it follow from this that each Banach space of density $\leqslant ...