
Real Analysis Real Analysis Math Forum 
 LinkBack  Thread Tools  Display Modes 
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 HahnBanach

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 
Senior Member Joined: Sep 2016 From: USA Posts: 647 Thanks: 411 Math Focus: Dynamical systems, analytic function theory, numerics  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, $d(x,S) = \sup\{\left \leftxy \right \right: y \in S\}$. 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  
Newbie Joined: Nov 2018 From: Canada Posts: 4 Thanks: 0  Quote:
Actually, it doesn't say anywhere in the exercise that its Banach spaces....  

Tags 
adherence, aherence, kernel, normed, space, subset, vector 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
kernel function  mhhojati  Linear Algebra  1  January 3rd, 2016 10:35 AM 
Denumerability of a subset of a denumerable subset  bschiavo  Real Analysis  9  October 6th, 2015 11:17 AM 
Subset of a Function g[a] subset g[b]  redgirl43  Applied Math  1  April 21st, 2013 06:20 AM 
Kernel of a function  c.P.u1  Linear Algebra  1  January 6th, 2011 06:43 AM 
The Kernel...  DanielThrice  Abstract Algebra  3  December 20th, 2010 03:03 PM 