# Function Spaces

• Nov 8th 2011, 10:34 AM
bugatti79
Function Spaces
Folks,

a) Does the function space $|| ||:C[0,1] \rightarrow R$ defined by $||f||=|f(1)-f(0)|$define a norm on C[0,1]

b)If it does, prove all the axioms for a norm hold. If not, demonstrate by an example some axiom which fails

How do I start a)....I have no idea....?

THanks
• Nov 8th 2011, 11:00 AM
Deveno
Re: Function Spaces
let $f(x) = x^2 - x$. then ||f|| = 0, but f ≠ 0.
• Nov 8th 2011, 11:39 AM
bugatti79
Re: Function Spaces
Quote:

Originally Posted by Deveno
let $f(x) = x^2 - x$. then ||f|| = 0, but f ≠ 0.

What I have in my notes is that a norm on a vector space V is a function || ||:V to R satisfying x and y in V and for all scalars alpha. One of the axioms states that ||x||=0 IFF x vector = zero vector.

Hence the above function does not define a norm...not sure how to rigorously prove it though...?

THanks
• Nov 8th 2011, 12:11 PM
TheEmptySet
Re: Function Spaces
Quote:

Originally Posted by bugatti79
What I have in my notes is that a norm on a vector space V is a function || ||:V to R satisfying x and y in V and for all scalars alpha. One of the axioms states that ||x||=0 IFF x vector = zero vector.

Hence the above function does not define a norm...not sure how to rigorously prove it though...?

THanks

To show something is false you only need to provide one counter example as Deveno did.
• Nov 8th 2011, 12:16 PM
bugatti79
Re: Function Spaces
Quote:

Originally Posted by TheEmptySet
To show something is false you only need to provide one counter example as Deveno did.

So thats the answer! Ok, just one other question...where is the rule that that a function cannot be = 0?

Thanks
• Nov 8th 2011, 12:21 PM
TheEmptySet
Re: Function Spaces
Quote:

Originally Posted by bugatti79
So thats the answer! Ok, just one other question...where is the rule that that a function cannot be = 0?

Thanks

A function can equal zero, but one of the axioms of a Norm is that the only objected that can have norm 0 is the zero of the space.

The function $f(x)=0$ is the zero of $C[0,1]$ Deveno's example gives another function that is not 0 that has lenth zero.
• Nov 8th 2011, 12:52 PM
bugatti79
Re: Function Spaces
Quote:

Originally Posted by TheEmptySet
A function can equal zero, but one of the axioms of a Norm is that the only objected that can have norm 0 is the zero of the space.

The function $f(x)=0$ is the zero of $C[0,1]$ Deveno's example gives another function that is not 0 that has lenth zero.

Ok, just to make sure I understand correctly. The given function does not define a norm on C[0,1] because the function itself must be 'zero' Ie, f is 0...?
• Nov 8th 2011, 12:57 PM
TheEmptySet
Re: Function Spaces
Quote:

Originally Posted by bugatti79
Ok, just to make sure I understand correctly. The given function does not define a norm on C[0,1] because the function itself must be 'zero' Ie, f is 0...?

Yes one of the properties of a norm is $||f||=0$ iff f is the zero of the space. Otherwise you get a psudonorm if that is the only property that fails.