It is currently Fri Feb 22, 2019 8:24 pm

 All times are UTC [ DST ]

 Page 1 of 1 [ 3 posts ]
 Print view Previous topic | Next topic
Author Message
 Post subject: The set of all polynomialsPosted: 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

 Post subject: Re: The set of all polynomialsPosted: 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

 Post subject: Re: The set of all polynomialsPosted: 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

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 1 [ 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 forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forumYou cannot post attachments in this forum

Search for:
 Jump to:  Select a forum ------------------ Algebra    Linear Algebra    Algebraic Structures    Homological Algebra Analysis    Real Analysis    Complex Analysis    Calculus    Multivariate Calculus    Functional Analysis    Measure and Integration Theory Geometry    Euclidean Geometry    Analytic Geometry    Projective Geometry, Solid Geometry    Differential Geometry Topology    General Topology    Algebraic Topology Category theory Algebraic Geometry Number theory Differential Equations    ODE    PDE Probability & Statistics Combinatorics General Mathematics Foundation Competitions Archives LaTeX    LaTeX & Mathjax    LaTeX code testings Meta