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?