Problem:

Let $\displaystyle X = C([0,1 ], R) $. Findd(f,g)for each of the following pairs of functions:

a. $\displaystyle f(x) =x $ and $\displaystyle g(x)=cos(x)$ for eachxin [0, 1]

b. I did not bother to write down this part; I just need to understand the notation properly.

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Definition:

$\displaystyle d(p,q) = \sqrt{\sum_{i=1}^n (p_{i}-q{i} )^2} $

So for part (a), I let $\displaystyle p_{i} = x $ and $\displaystyle q_{i}=cosx$ and n = 1, since this is in $\displaystyle \mathbb{R}$.

$\displaystyle d(f, g) = \sqrt{\sum_{i=1}^1 (x - cos(x))^2} $

but here I am confused. Would the d(f, g) be

$\displaystyle d(f, g) = \sqrt{(1-cos(1))^2 + (0 - cos(0))^2} $ ?

Thank you for your help. Much appreciated.