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 November 25th, 2018, 11:57 AM #1 Newbie   Joined: Nov 2018 From: Canada Posts: 4 Thanks: 0 Adherence of subset - Kernel Having Y, a subspace of X. How can we show that the adherence of Y can be expressed as : Adherence of Y = intersection of { Ker(f) | f element of X* , Y contained in Ker(f)} November 25th, 2018, 12:07 PM #2 Senior Member   Joined: Oct 2009 Posts: 867 Thanks: 330 Use Hahn-Banach November 25th, 2018, 12:22 PM #3 Newbie   Joined: Nov 2018 From: Canada Posts: 4 Thanks: 0 Can you give more details please ?  November 25th, 2018, 03:09 PM   #4
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 Originally Posted by whatsup123 Can you give more details please ? I'm assuming you are given that $X,Y$ are Banach spaces yes? Then fix $x_0 \in X \setminus \text{cl}(Y)$ and define a subspace, $X_0 = Y \bigcup \text{span}(x_0)$ and a linear functional, $x_0^* \in X_0^*$ by
$x_0^*(x) = \frac{d(x,Y)}{d(x_0,Y)}$
where $d: X \to \mathbb{R}$ is the set distance, $d(x,S) = \sup\{\left| \left|x-y \right| \right|: y \in S\}$.

Now, show that you can extend $x_0^*$ to a linear functional on the entire space and that this extension is a competitor in your intersection since $Y \in \text{ker}(x_0^*)$. November 25th, 2018, 03:19 PM   #5
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Quote:
 Originally Posted by SDK I'm assuming you are given that $X,Y$ are Banach spaces yes? Then fix $x_0 \in X \setminus \text{cl}(Y)$ and define a subspace, $X_0 = Y \bigcup \text{span}(x_0)$ and a linear functional, $x_0^* \in X_0^*$ by $x_0^*(x) = \frac{d(x,Y)}{d(x_0,Y)}$ where $d: X \to \mathbb{R}$ is the set distance, $d(x,S) = \sup\{\left| \left|x-y \right| \right|: y \in S\}$. Now, show that you can extend $x_0^*$ to a linear functional on the entire space and that this extension is a competitor in your intersection since $Y \in \text{ker}(x_0^*)$.

Actually, it doesn't say anywhere in the exercise that its Banach spaces.... Tags adherence, aherence, kernel, normed, space, subset, vector Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post mhhojati Linear Algebra 1 January 3rd, 2016 10:35 AM bschiavo Real Analysis 9 October 6th, 2015 11:17 AM redgirl43 Applied Math 1 April 21st, 2013 06:20 AM c.P.u1 Linear Algebra 1 January 6th, 2011 06:43 AM DanielThrice Abstract Algebra 3 December 20th, 2010 03:03 PM

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