Let X=C[0,1]. For f,g in X, define
d(f,g) = sup(x 0 to 1)|f(x)-g(x)|. Calculate the distance between f(x)=x and g(x) = x^2

Question

Let X=C[0,1]. For f,g in X, define 
d(f,g) = sup(x 0 to 1)|f(x)-g(x)|. Calculate the distance between f(x)=x and g(x) = x^2

experimentx · Answer

isn't it maximizing this function |x - x^2| in the interval 0 to 1

anonymous · Answer

not sure..my lectrer said find the max of f-g first..

experimentx · Answer

this is just a parabola
x(x-1) .. vertex is at 0.5

experimentx · Answer

can you define d(f,g) again?

anonymous · Answer

d(f,g)=$$\sup _{0\le x \le 1} |f(x)-g(x)|$$

experimentx · Answer

that's definitely the supremum of |f(x) - g(x)| in the interval 0-2 ...it must be 0.5

experimentx · Answer

http://www.wolframalpha.com/input/?i=plot+x+-+x^2

experimentx · Answer

* interval 0-1

anonymous · Answer

means that the distance is the max value of f-g?and also the supremum?

anonymous · Answer

It's the maximum of the absolute value.  So it could be the minimum if it is further away from 0.  Since 0.5 is where the max is, you plug it in to find the distance: (0.5)(1-0.5)=0.25

anonymous · Answer

oo..ok.thanks