My Math Forum  

Go Back   My Math Forum > College Math Forum > Real Analysis

Real Analysis Real Analysis Math Forum


Reply
 
LinkBack Thread Tools Display Modes
March 12th, 2013, 07:02 AM   #1
Newbie
 
Joined: Mar 2013

Posts: 16
Thanks: 0

Minkowski Functional used in Proof Hahn Banach Theorem

We take X to be a normed vector space.
We define the Minkowski Functional as: where C is an open convex subset of X, containing the origin.

The property I want to look at is:

, where M is some constant.

Please view attachment for proof of this property.

Can you see the reasoning of how we can state in the last line of the attached proof.

Thanks
Attached Images
File Type: jpg Proof of property.jpg (24.7 KB, 392 views)
PeterPan is offline  
 
March 12th, 2013, 08:41 AM   #2
Senior Member
 
Joined: Feb 2013

Posts: 281
Thanks: 0

Re: Minkowski Functional used in Proof Hahn Banach Theorem

That vector is an element of C, becuase it is an element of B(0, r) and B < C. Convince yourself that it's an element of B(0, r) indeed, taking it's norm and showing this norm is less than r.

To understand the statement deeper it's useful to think about our real space. C is a potato around the origin. B is a ball inside the potato with radii r. Obviously you can shrink any x vector to the ball and r*x/|x| is on the ball's surface.
csak is offline  
March 12th, 2013, 09:46 AM   #3
Newbie
 
Joined: Mar 2013

Posts: 16
Thanks: 0

Re: Minkowski Functional used in Proof Hahn Banach Theorem

Yes I see if I take the norm on both sides then
Using your geometric interpretation do you imagine the vector x as being an arrow extending from the origin and enclosed by the open ball?
PeterPan is offline  
March 12th, 2013, 10:19 AM   #4
Senior Member
 
Joined: Feb 2013

Posts: 281
Thanks: 0

Re: Minkowski Functional used in Proof Hahn Banach Theorem

Yes, when we think about as a vector space, then we interpret x = (x1 ,x2 ,x3) as a vector arrow from the origin pointing to (x1, x2, x3). The beginning of the arrow is fixed at the origin.
csak is offline  
Reply

  My Math Forum > College Math Forum > Real Analysis

Tags
banach, functional, hahn, minkowski, proof, theorem



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Routine proof in Functional Analysis AfroMike Real Analysis 6 August 28th, 2013 01:23 PM
Routine proof in Functional Analysis AfroMike Real Analysis 1 August 21st, 2013 11:57 AM
Using Hahn-Banach Geometric Form Kramer Real Analysis 1 May 6th, 2013 03:14 AM
Kakutani Theorem (functional analysis) PeterPan Real Analysis 1 April 29th, 2013 12:46 PM
Minkowski's theorem lotte Number Theory 0 July 28th, 2010 06:29 AM





Copyright © 2017 My Math Forum. All rights reserved.