My Math Forum Minkowski Functional used in Proof Hahn Banach Theorem

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March 12th, 2013, 07:02 AM   #1
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Minkowski Functional used in Proof Hahn Banach Theorem

We take X to be a normed vector space.
We define the Minkowski Functional as: $p(x) := \text{inf}\{ t > 0 : \ \frac{x}{t} \in C\}=$ where C is an open convex subset of X, containing the origin.

The property I want to look at is:

$0 \leq p(x) \leq M||x||_{X} \text{ for all } x \in X$ , where M is some constant.

Please view attachment for proof of this property.

Can you see the reasoning of how we can state $\frac{x}{||x||_{X}}\frac{r}{2} \in C$ in the last line of the attached proof.

Thanks
Attached Images
 Proof of property.jpg (24.7 KB, 392 views)

 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.
 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 $\frac{r}{2}\frac{||x||}{||x||} \= \frac{r}{2} \ < \ r=$ Using your geometric interpretation do you imagine the vector x as being an arrow extending from the origin and enclosed by the open ball?
 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 $^\mathbb{R}^3$ 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.

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

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