Hello

It is $\displaystyle <u,v> = \int^2_0 \overline{u}(x) v(x) dx \ \ \forall u,v \in C[0,2] $ (defines a dot product)

Show that $\displaystyle f_n$ is a cauchy-sequenze

(field) norm: $\displaystyle ||u||_2 = \sqrt{<u, u>} \ \ \forall u \in C[0,2]$

$\displaystyle f_n(x)=\begin{cases} 0, & 0\le x\le 1- \frac{1}{n} \\ n(x-(1-\frac{1}{n})), & 1-\frac{1}{n}<x<1\\ 1, & 1\le x \le 2 \end{cases}$

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I think $\displaystyle \lim_{n \to \infty} <f_{n+1}, f_n> = 0 $is okay, but it is

$\displaystyle \lim_{n \to \infty} <f_{n+1}, f_n> = \infty$

any help would be really apprectiated

Rapha