Thread: Function Spaces

Folks,

a) Does the function space defined by 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

let . then ||f|| = 0, but f ≠ 0.

Originally Posted by Deveno

let . 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

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.

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

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 is the zero of Deveno's example gives another function that is not 0 that has lenth zero.

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 is the zero of 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...?

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 iff f is the zero of the space. Otherwise you get a psudonorm if that is the only property that fails.