# Convergence in space of all square integrable functions on [-1,1]

• Feb 11th 2013, 10:10 PM
Mollier
Convergence in space of all square integrable functions on [-1,1]
Problem:
1. Given the sequence $\displaystyle x_n(s) = (|s|+\frac{1}{n})^{-1/3}$ and function $\displaystyle x(s)=|s|^{-1/3}$. show that $\displaystyle x_n\rightarrow x$
in $\displaystyle L_2(-1,1)$

2. How can we use this to deduce that $\displaystyle C[-1,1]$ (with the max norm) is not complete.

Pathetic Attempt:

1. $\displaystyle lim_{n\rightarrow \infty}||x_n(s)-x(s)||_{L_2} = lim_{n\rightarrow \infty} \int^1_{-1} |x_n(s)-x(s)|^2 = 0$
because $\displaystyle lim_{n\rightarrow \infty} 1/n = 0$.

2. The limit $\displaystyle x(s)$ isn't in C[-1,1] so its not complete...

Any suggestions are welcome.
• Feb 12th 2013, 11:51 AM
Mollier
Re: Convergence in space of all square integrable functions on [-1,1]
A somewhat better attempt:

$\displaystyle \int^1_{-1} |x_n(s) - x(s)|^2 ds$

I split the integral up in two for simplicity.

$\displaystyle \int^1_{-1} (|s|+1/n)^{-1/3}ds = 6(1/n+1)^{1/3} - 6(1/n)^{1/3}$

$\displaystyle \int^1_{-1} |s|^{-1/3}ds = 6$

Let n go to infinity and we have proved convergence.

Still working on the other part though.