completion of space

**hermanni** · February 20th 2011, 07:29 AM

Hi all,
I can't solve the following , can anyone help?
Let X = { $(x_{n})_{n=1}^{\infty} : \sum_{n=1}^{\infty} n^{-2}| x_{n}| < \infty$ } , and a norm on this space is defined by : || x || = $sup_{n > 0} |x_{n}| n^{-2}$ Show that the space (X , || .||) is not complete and find its completion. Obviously I need to find a complete and non-convergent sequence in this space , can anyone give an idea ? For the completion , I don't have an idea

any hint is appreciated

**Opalg** · February 20th 2011, 10:10 AM

Originally Posted by hermanni

Hi all,
I can't solve the following , can anyone help?
Let X = { $(x_{n})_{n=1}^{\infty} : \sum_{n=1}^{\infty} n^{-2}| x_{n}| < \infty$ } , and a norm on this space is defined by : || x || = $\sup_{n > 0} |x_{n}| n^{-2}$ Show that the space (X , || .||) is not complete and find its completion. Obviously I need to find a complete and non-convergent sequence in this space , can anyone give an idea ? For the completion , I don't have an idea

any hint is appreciated

For a Cauchy sequence that does not converge, take a look at $(x^{(n)})$ , where $x^{(n)} = (1,2,3,\ldots,n,0,0,0,\ldots).$

For the completion, you should expect it to be a space in which the formula for a point to belong to the space coincides with the formula for the norm. So I would expect the completion to be the space $\{(x_{n})_{n=1}^{\infty} : \sup_{n>0} n^{-2}| x_{n}| < \infty \}.$

Edit. No, I was wrong about the completion of the space. I think it should be $\{(x_{n})_{n=1}^{\infty} : \lim_{n\to\infty} n^{-2} x_{n} = 0 \}.$ That is complete in the given norm, and it contains $X$ as a dense subspace.

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