In $\mathcal{C}([0,1])$ find $d(f,g)$ where $f(x) = 1$ and $g(x) = x$ for both the $\sup$ norm and the norm given in Example 1.7.7. (Here we mean the induced norm from a given inner-product, and its associated induced metric.)
The example 1.7.7, assumes that inner product is $\langle f,g\rangle = \int_0^1 f(x)g(x) \ dx $ on ${C}([0,1])$
For the first one, because the largest distance between $f,g$ are $1-0=1$
I am having trouble figuring out the answer.
The answer is $1$ and $1/ \sqrt3$.
$d(f,g)=\sqrt {\int_0^{1} (1-x)^{2}dx} =\sqrt {\frac 1 3}$ for the second metric.