Norm of a Function vs. Length of a Vector

[pr0me7heu2]

Suppose f(x)= -2x+1 is a vector in the vector space C[0,1].

Calculating the norm (f,f) results in 1/3.

I'm a little confused.

So on [0,1] the function is a straight line from (0,1) to (0,-1).

So I thought I could simply takes this line segment and turn it into a directed line segment originating from the origin. So it would be equivalent to the vector v= 0i - 2j (right?)

, so then ||v|| = sqr(0^2 + (-2)^2) = 2

So the length of vector v is 2.

Why is this different from the norm (f,f)? Shouldn't they be the same?

...or am I completely missing the point here of the norm / inner product of the function?

 

[John Creighto]

Suppose f(x)= -2x+1 is a vector in the vector space C[0,1].

Calculating the norm (f,f) results in 1/3.

I'm a little confused.

So on [0,1] the function is a straight line from (0,1) to (0,-1).

So I thought I could simply takes this line segment and turn it into a directed line segment originating from the origin. So it would be equivalent to the vector v= 0i - 2j (right?)

, so then ||v|| = sqr(0^2 + (-2)^2) = 2

So the length of vector v is 2.

Why is this different from the norm (f,f)? Shouldn't they be the same?

...or am I completely missing the point here of the norm / inner product of the function?

There is more then one definition for the norm of a function.

 

[HallsofIvy]

The length of a curve (or straight line) has little to do with its "norm". Why do you think they should be the same?

Your title, "Norm of a Vector Versus Length of a Vector" is somewhat misleading. You are actually talking about the norm of a function (thought of as a vector) and the length of its graph which is not at all a vector.

 

[floyd13]

The norm of the function defined in this case is (f.f) = $$\oint f^2 dx$$
The limits are from 0 to 1.
The above integral turns out to be 1/3 which is the correct answer.