Exercise 4 (Uniform boundedness principle-General case). Suppose X is a Baire space. 1) Consider a family of continuous functions ; : X → R, i I. Suppose that supp(x) < +∞o, for every x € X. iЄI Show that there exists finite constant K and a non-empty open set OC X such that supp:(x)| ≤ K, for every x ≤ 0.
Exercise 4 (Uniform boundedness principle-General case). Suppose X is a Baire space. 1) Consider a family of continuous functions ; : X → R, i I. Suppose that supp(x) < +∞o, for every x € X. iЄI Show that there exists finite constant K and a non-empty open set OC X such that supp:(x)| ≤ K, for every x ≤ 0.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Exercise 4
Part 1
![**Exercise 4 (Uniform boundedness principle-General case).**
Suppose \( X \) is a Baire space.
1) Consider a family of continuous functions \( \varphi_i : X \rightarrow \mathbb{R}, i \in I \). Suppose that
\[
\sup_{i \in I} |\varphi_i(x)| < +\infty, \text{ for every } x \in X.
\]
Show that there exists a finite constant \( K \) and a non-empty open set \( O \subset X \) such that
\[
\sup_{i \in I} |\varphi_i(x)| \leq K, \text{ for every } x \in O.
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Transcribed Image Text:**Exercise 4 (Uniform boundedness principle-General case).**
Suppose \( X \) is a Baire space.
1) Consider a family of continuous functions \( \varphi_i : X \rightarrow \mathbb{R}, i \in I \). Suppose that
\[
\sup_{i \in I} |\varphi_i(x)| < +\infty, \text{ for every } x \in X.
\]
Show that there exists a finite constant \( K \) and a non-empty open set \( O \subset X \) such that
\[
\sup_{i \in I} |\varphi_i(x)| \leq K, \text{ for every } x \in O.
\]
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