Dense inclusions of Banach spaces and their duals - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T22:33:34Z http://mathoverflow.net/feeds/question/14960 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/14960/dense-inclusions-of-banach-spaces-and-their-duals Dense inclusions of Banach spaces and their duals Tom LaGatta 2010-02-11T01:55:26Z 2010-02-11T02:11:00Z <p>This seems like a really simple question, but I'm struggling with it. Let $X$ be a separable Banach space, $H$ be a separable Hilbert space, and suppose $i : H \hookrightarrow X$ is a dense, continuous embedding of $H$ into $X$. (This is the abstract Wiener space construction due to Gross, hence the [pr.probability] tag) If we associate $H$ with its dual $H^*$, we have the inclusions $$X^* \hookrightarrow H^* \cong H \hookrightarrow X.$$</p> <p><b>My question:</b> Is $i^* : X^* \hookrightarrow H^*$ a dense injection?</p> http://mathoverflow.net/questions/14960/dense-inclusions-of-banach-spaces-and-their-duals/14963#14963 Answer by Bill Johnson for Dense inclusions of Banach spaces and their duals Bill Johnson 2010-02-11T02:11:00Z 2010-02-11T02:11:00Z <p>Yes, if you mean that $i$ is one to one, for an operator $T:X\to Y$ is one to one if and only if $T$* has weak* dense range, which means $T$* has dense range when $X$ is reflexive.</p>