Is U an Open Subset in the Space of Continuous Functions?

[Jonmundsson]

Homework Statement

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

Homework Equations

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

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

$\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$

The Attempt at a Solution

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

$\displaystyle |\int _a ^b f(x)dx| \leq \int _a ^b |f(x)|dx \leq \int _a ^b ||f||_I dx = ||f||_I \int _a ^b dx = (b-a)||f||_I$

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.

 

[mighty2000]

Hello there,

Yes, your approach is correct. To show that U is open, we need to show that for any continuous function f in U, there exists an open ball around f in \mathcal{C}_{\mathbb{K}}(I) that is contained in U. In other words, we need to show that for any f in U, there exists an \epsilon > 0 such that all continuous functions g in \mathcal{C}_{\mathbb{K}}(I) with ||f-g||_I < \epsilon are also in U.

The purpose of the norm in this problem is to measure the distance between two functions in \mathcal{C}_{\mathbb{K}}(I). By using the supremum norm, we are essentially looking at the maximum difference between the values of the two functions over the interval I. This allows us to control the error in our approximation of the integral \displaystyle \int _a ^b f(x)dx.

I hope this helps. Let me know if you have any other questions or need further clarification.