Assume that $X,Y$ are infinite dimensional Banach spaces. Is it true that if density character of $X$ is less then or equal to density character of $Y$ then $card X \leq card (Y)$ ?



Density character and cardinality
Let $\aleph$ be the density character of the Banach space $X$ and let's compute the cardinality of $X$ using AC but not much set theory, since these days cardinal arithmetic is generally given short shrift in real analysis courses.


Take a dense set $D=\{x\_a: a<\aleph\}$ in $X$ and observe that all tails of this set are still dense, so for each $x$ in $X$ there is a sequence $x\_{a\_0}<x\_{a\_1} <\dots$ in $D$ that converges to $x$. This gives an upper bound for the cardinality of $X$. 


To get the corresponding lower bound, take a set $\{y\_a: a<\aleph\}$ of unit vectors in $D$ so that for each $a < \aleph$, the distance of $y\_a$ to the span of $\{y\_b: b<a\}$ is larger than $1/2$. Given $A=\{a\_0<a\_1<\dots<\aleph\}$ define $z\_A = \sum\_{n=0}^\infty 10^{-n} y\_{a\_n}$. It remains to show that if $A\not=B$ then $z\_A\not= z\_B$. 


Write $A=\{a\_0<a\_1<\dots<\aleph\}$ and $B=\{b\_0<b\_1<\dots<\aleph\}$. If $a\_0\not= b\_0$, then $\|y\_{a\_0}-y\_{b\_0}\|> 1/2$ and the desired result follows from the triangle inequality. In the general case let $m$ be the smallest $n$ s.t. $y\_{a\_n}\not= y\_{b\_n}$. Apply the first case to 
$10^m\sum\_{n=m}^\infty 10^{-n} y\_{a\_n}$ and 
$10^m\sum\_{n=m}^\infty 10^{-n} y\_{b\_n}$.


