A question about density character of Banach spaces. - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-23T20:58:27Z http://mathoverflow.net/feeds/question/79662 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79662/a-question-about-density-character-of-banach-spaces A question about density character of Banach spaces. Peter 2011-10-31T22:51:13Z 2011-11-03T21:28:11Z <p>Let $\langle M_i:i&lt;\theta\rangle$ be an increasing chain of Banach spaces, where each $M_i$ has density character $\mu$ (i.e.,the mininum cardinality of a dense subset of $M_i$ is $\mu$). Let $B_i\subset M_i$ be a dense subset of $M_i$ of cardinality $B_i$. Notice that $\bigcup_{i&lt;\theta}B_i$ is a dense subset of $\overline{\bigcup_{i&lt;\theta}M_i}=:M$, so $density-character(M)\le \mu$. Is it possible to prove that $density-character(M)=\mu$? Thank you.</p> http://mathoverflow.net/questions/79662/a-question-about-density-character-of-banach-spaces/79982#79982 Answer by Ramiro de la Vega for A question about density character of Banach spaces. Ramiro de la Vega 2011-11-03T21:28:11Z 2011-11-03T21:28:11Z <p>If $X$ is any metric space and $Y$ is any subspace of $X$ then $dc(Y) \leq dc(X)$. </p>