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Justin
 Post subject: Space of Continuous Functions
PostPosted: Mon, 23 Jan 2012 15:54:46 UTC 
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Reviewing my analysis here:

Let B=\{f \in C_b(\mathbb{R}, \mathbb{R})|f(x)>0, \forall x \in \mathbb{R}\}.
I want to prove that B is not open. The most obvious approach for me is to show that the interior of B is a proper subset, i.e., \text{int}(B) \subset B.
I'm just having trouble coming up with an example function.
I've been trying variants of the exponential function such as e^{-x^2} and it's not working.
I know I can write \text{int}(B)=\{f \in C_b(\mathbb{R}, \mathbb{R})| \exists \delta > 0, \text{such that} , f(x) > \delta , \forall x \in \mathbb{R}\}.

Any ideas? :?

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"In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy."
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Shadow
 Post subject: Re: Space of Continuous Functions
PostPosted: Mon, 23 Jan 2012 17:22:16 UTC 
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Justin wrote:
Reviewing my analysis here:

Let B=\{f \in C_b(\mathbb{R}, \mathbb{R})|f(x)>0, \forall x \in \mathbb{R}\}.
I want to prove that B is not open. The most obvious approach for me is to show that the interior of B is a proper subset, i.e., \text{int}(B) \subset B.
I'm just having trouble coming up with an example function.
I've been trying variants of the exponential function such as e^{-x^2} and it's not working.
I know I can write \text{int}(B)=\{f \in C_b(\mathbb{R}, \mathbb{R})| \exists \delta > 0, \text{such that} , f(x) > \delta , \forall x \in \mathbb{R}\}.

Any ideas? :?


What's the problem with the function f(x)= e^x? It's not in any of those sets which form the interior, since there is no such lower bound...

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Justin
 Post subject: Re: Space of Continuous Functions
PostPosted: Mon, 23 Jan 2012 18:51:19 UTC 
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Shadow wrote:
What's the problem with the function f(x)= e^x? It's not in any of those sets which form the interior, since there is no such lower bound...


Am I missing something? f(x) = e^x is not bounded on \mathbb{R}...

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"Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another." Bernard Le Bovier Fontenelle (1657-1757)

"In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy."
G.H. Hardy (1877-1947)
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Shadow
 Post subject: Re: Space of Continuous Functions
PostPosted: Mon, 23 Jan 2012 22:35:35 UTC 
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Justin wrote:
Shadow wrote:
What's the problem with the function f(x)= e^x? It's not in any of those sets which form the interior, since there is no such lower bound...


Am I missing something? f(x) = e^x is not bounded on \mathbb{R}...


Oh right, so e^{-x^2}

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Justin
 Post subject: Re: Space of Continuous Functions
PostPosted: Sun, 29 Jan 2012 20:04:04 UTC 
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Shadow wrote:
Justin wrote:
Shadow wrote:
What's the problem with the function f(x)= e^x? It's not in any of those sets which form the interior, since there is no such lower bound...


Am I missing something? f(x) = e^x is not bounded on \mathbb{R}...


Oh right, so e^{-x^2}


Yes, this turned out to work...I think I just needed someone to confirm my answer.

Thanks! :D

_________________
"Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another." Bernard Le Bovier Fontenelle (1657-1757)

"In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy."
G.H. Hardy (1877-1947)
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Shadow
 Post subject: Re: Space of Continuous Functions
PostPosted: Sun, 29 Jan 2012 22:27:56 UTC 
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
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Justin wrote:
Shadow wrote:
Justin wrote:
Shadow wrote:
What's the problem with the function f(x)= e^x? It's not in any of those sets which form the interior, since there is no such lower bound...


Am I missing something? f(x) = e^x is not bounded on \mathbb{R}...


Oh right, so e^{-x^2}


Yes, this turned out to work...I think I just needed someone to confirm my answer.

Thanks! :D


:)

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