Solution Found!
Show that the space of all continuous functions defined on
Chapter 4, Problem 28E(choose chapter or problem)
QUESTION:
Show that the space \(C(\mathbb{R})\) of all continuous functions defined on the real line is an infinite-dimensional space.
Questions & Answers
QUESTION:
Show that the space \(C(\mathbb{R})\) of all continuous functions defined on the real line is an infinite-dimensional space.
ANSWER:Solution 28E
Step 1 of 3
Assume that the space of all continuous functions.
Objective is to show that defined on the real line is an infinite dimensional space.
Let be the real vector space of all continuous real valued functions on the interval
For any value of the list is linearly independent in
To prove this suppose that there are, such that
For every
Thus, all the coefficients are zero; hence the set is linearly independent in
