I'm trying to prove that the normed space of all piecewise continuously functions with the norm $$\int^1_{-1}|f(x)|^2dx$$ is not a complete normed space.
$L_2PC[-1,1]$
for that, im trying to find a Cauchy sequence function which does not converge in this space with this norm. (maybe the Cauchy sequence converges, but to a function which doesn't belong to $L_2PC[-1,1]$)
I have been stuck on this for a while, and I coule use some help. Thanks
Consider the sequence $(f_n)_{n \geq 0}$ with $$ f_n(x) = \sum_{k=0}^n 2^{-k} \chi_{[1-2^{-k+1},\, 1-2^{-k})}(x) $$ where $\chi$ denotes the indicator function. If I got the indices right these should be step functions where $f_{n+1}$ differs to $f_n$ by a new step of length $2^{-(n+1)}$ and height $2^{-(n+1)}$. Every $f_n$ is piecewise continuous and they converge in $L^2$ to $$ f(x) = \sum_{k=0}^\infty 2^{-k} \chi_{[1-2^{-k+1},\, 1-2^{-k})}(x), $$ but this limit has an infinite number of discontinuities.