I am given a space with inner product $\int_{0}^{2}f(x)g(x)dx$ and told to find a vector of length 1. I know that the length of a vector is $\sqrt{a*a}$, so this one would be set up as $1=\sqrt{\int_0^2f(g)^2dx} $ which can be simplified to $1^2=\int_0^2f(g)^2dx$.
Is this set up correctly? If so, how do I go about solving this?
Thank you!
Diven any function $g:[0,2] \to \mathbb{R}$ with non-zero norm $\|g\|$, one can define the function $f [9,2] \to \mathbb{R}$ by $$ f(x)=\lambda g(x), $$ where $\lambda=1/\|g\|$.
Then $$ \|f\|=\|\lambda g\|=\lambda \||g\|=\frac{\||g\|}{\|g\|}=1 $$
For instance, if $g: [0.2] \to \mathbb{R}$ is defined by $g(x)=\sqrt{2-x}$, then $$ \|g\|=\sqrt{\int_0^2(2-x)dx}=\sqrt{[2x-0.5x^2]_0^2}=\sqrt{2} $$ The function $f:[0,2] \to \mathbb{R}$ defined by $f(x)=\sqrt{1-0.5x}$ will then have $1$ as norm .