Subspaces of functions

  • Thread starter gaborfk
  • Start date
53
0

Main Question or Discussion Point

Yet another problem I need to get some starting help on:

Show that the set of continuous functions f=f(x) on [a,b] such that [tex]\int \limits_a^b f(x) dx=0[/tex] is a subspace of C[a,b]
Thank you
 

Answers and Replies

NateTG
Science Advisor
Homework Helper
2,448
5
I would start by checking the definition of subspace.
 
53
0
Definition of subspace means that the functions are closed under addition and scalar multiplication
 
NateTG
Science Advisor
Homework Helper
2,448
5
gaborfk said:
Definition of subspace means that the functions are closed under addition and scalar multiplication
So can you show that that's true for the potential subspace in your example?
 
53
0
You mean that if [tex]\int \limits_a^b f(x) dx=0[/tex] and [tex]\int \limits_a^b g(x) dx=0[/tex], can I prove that [tex]\int \limits_a^b f(x)+g(x) dx=0[/tex]? Also, if [tex]\int \limits_a^b f(x) dx=0[/tex] then [tex]k\int \limits_a^b f(x) dx=0[/tex]?
 
NateTG
Science Advisor
Homework Helper
2,448
5
Yeah, that's pretty much it. (Technically you also have to show that it's a subset, but in this case that's trivial.)
 
53
0
Thank you!

The "hard ones" are so easy sometimes.....
 

Related Threads for: Subspaces of functions

  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
2
Views
8K
  • Last Post
Replies
9
Views
2K
  • Last Post
Replies
5
Views
7K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
17
Views
4K
  • Last Post
Replies
10
Views
6K
Top