1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Metric Spaces - Open subset

  1. Feb 5, 2012 #1
    1. The problem statement, all variables and given/known data
    Let [itex]a,b \in \mathbb{R}[/itex] and we define [itex]I = [a,b][/itex] ([] means closed set). Let [itex]\mathcal{C}_{\mathbb{K}}(I)[/itex] be the space of all continuous functions [itex]I \to \mathbb{K}[/itex] with the norm [itex]f \mapsto ||f||_I = \displaystyle \sup _{x \in I} |f(x)|[/itex]. Let [itex]U[/itex] be the set of all continuous functions [itex]I \to \mathbb{K}[/itex] so that [itex]\displaystyle |\int _a ^b f(x)dx| < 1[/itex]. Show that [itex]U[/itex] is an open subset of [itex]\mathcal{C}_{\mathbb{K}}(I)[/itex].


    2. Relevant equations
    If [itex]f: X \to Y[/itex] is a continuous function and [itex]B \subset Y[/itex] is open then the preimage [itex]f^{-1}[/itex] is open.

    For [itex]B \subset Y[/itex] the set [itex]f^{-1} = \{x \in X : f(x) \in B\}[/itex] is called the preimage of B.

    [itex]\mathbb{K} = \mathbb{R}[/itex] or [itex]\mathbb{C}[/itex]

    3. The attempt at a solution

    The function [itex]\mathcal{C} _{\mathbb{K}} (I) \to \mathbb{K}, f \mapsto \displaystyle \int _a ^b f(x)dx[/itex] is continuous because

    [itex] \displaystyle |\int _a ^b f(x)dx| \leq \int _a ^b |f(x)|dx \leq \int _a ^b ||f||_I dx = ||f||_I \int _a ^b dx = (b-a)||f||_I[/itex]

    I want to show that preimage of U is open then U is open using the two definitions I listed above. I'm just curious whether that is a legit method to solve this problem. I'm also wondering what the purpose of the norm is in this problem. If anyone can answer these questions I will be grateful.

    Thanks.
     
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Loading...