- #1

Lily@pie

- 109

- 0

## Homework Statement

The vector space of continuous functions on [0,1] is given the following metrics

[itex] d_{∞} (f,g) = sup_{x\in [0,1]} |f(x)-g(x)| [/itex]

[itex] d_{2} (f,g) = (∫^{1}_{0} |f(x)-g(x)|^{2} dx)^{1/2}[/itex]

Are these two metrics equivalent?

## Homework Equations

If [itex] d_{∞} [/itex] and [itex] d_{2} [/itex] are equivalent, then for every [itex]f \in C[0,1][/itex] and every [itex]\epsilon>0[/itex], there exist a [itex]\delta>0[/itex] such that [itex]B^{d_{∞}}_{\delta}(f)\subset B^{d_{2}}_{\epsilon}(f)[/itex] and [itex]B^{d_{2}}_{\delta}(f)\subset B^{d_{∞}}_{\epsilon}(f)[/itex]

## The Attempt at a Solution

I know that these two metrics are not equivalent, but i can't find a function at which i can find an [itex]\epsilon>0[/itex] for every [itex]\delta>0[/itex] where they are not the subset of each other.

I know that [itex] d_{∞} [/itex] is like the highest point of the g(x) if I let f(x)=0 and g(x) be some other function. [itex] d_{2} [/itex] is like the area under the graph between [0,1].