It is currently Tue Dec 18, 2018 11:18 pm


All times are UTC [ DST ]




Post new topic Reply to topic  [ 3 posts ] 
Author Message
PostPosted: Wed May 17, 2017 2:12 pm 
Team Member

Joined: Mon Nov 09, 2015 1:52 pm
Posts: 426
Is the set of all polynomials open in \(\displaystyle{\left(C([-1,1])\,,||\cdot||_{\infty}\right)}\) ?


Top
Offline Profile  
Reply with quote  

PostPosted: Thu May 18, 2017 7:44 pm 

Joined: Thu Dec 10, 2015 1:58 pm
Posts: 59
Location: India
Polynomials being a subspace of Banach Space $(C[-1,1],\lVert \cdot \rVert_{\infty})$, must have empty interior, otherwise it'd have to be the full space.

(One argument for showing it's not closed, Polynomials have a countable Hamel basis and Banach spaces of infinite dimension cannot have a countable Hamel basis).


Top
Offline Profile  
Reply with quote  

PostPosted: Thu May 18, 2017 8:47 pm 
Team Member

Joined: Mon Nov 09, 2015 1:52 pm
Posts: 426
Thank you r9m.

Here is another idea.

Suppose that \(\displaystyle{\mathcal{P}}\) (the set of polynomials) is open in \(\displaystyle{\left(C([-1,1]),||\cdot||_{\infty}\right)}\).

Let \(\displaystyle{f(x)=x\,,x\in\left[-1,1\right]}\) and then \(\displaystyle{f\in\mathcal{P}\subseteq C([-1,1])}\).

There exists \(\displaystyle{\epsilon>0}\) such that \(\displaystyle{B(f,\epsilon)\subseteq \mathcal{P}}\).

Consider the continuous function \(\displaystyle{g(x)=f(x)+\dfrac{\epsilon\,x}{2\,(1+|x|)}\,,x\in\left[-1,1\right]}\)

Observe that \(\displaystyle{\forall\,x\in\left[-1,1\right]\,\,,|g(x)-f(x)|=\dfrac{\epsilon\,|x|}{2\,(1+|x|)}<\dfrac{\epsilon}{2}}\)

so, \(\displaystyle{||g-f||_{\infty}\leq \dfrac{\epsilon}{2}<\epsilon\implies g\in B(f,\epsilon)\implies g\in\mathcal{P}}\)

which is a contradiction since \(\displaystyle{g}\) is not twice differentiable at \(\displaystyle{x=0}\).


Top
Offline Profile  
Reply with quote  

Display posts from previous:  Sort by  
Post new topic Reply to topic  [ 3 posts ] 

All times are UTC [ DST ]


Mathimatikoi Online

Users browsing this forum: No registered users and 1 guest


You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot post attachments in this forum

Search for:
Jump to:  
Powered by phpBB® Forum Software © phpBB Group Color scheme created with Colorize It.
Theme created StylerBB.net