How to prove the metric space $L^{p}[a,b]$ is a complete metric space using the definition that says, Every Cauchy sequence in the metric space should converge to some point in that space? $$\left\{x(t)\in C[a,b] : \int_a^b |x(t)|^p \;dt < \infty \right\};$$ $$||x(t)||=\left[\int_a^b|x(t)|^p \;dt\right]^\frac{1}{p}$$
How to prove this metric space is a complete metric space?
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$\begingroup$
functional-analysis
banach-spaces
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0Is it a homework problem? What did you try? – 2011-12-19
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3Your description of $L^p[a,b]$ is inaccurate. The space $L^p[a,b]$ of [$p$-integrable functions](http://en.wikipedia.org/wiki/Lp_space)is not the space of continuous function with finite $L^p$-norm. By the way: the result you ask about is called the [Riesz-Fischer theorem](http://en.wikipedia.org/wiki/Riesz-Fischer_theorem) – 2011-12-19
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0Furthermore, your notation of the norm of $x$ is ugly. Maybe you forgot to take the closure of your space. – 2011-12-19
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0In that case it is complete by definition. – 2011-12-19
2 Answers
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Have you tried searching the internet. These are standard questions in functional analysis. You can find an answer at the below link.
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0thank you for your response. – 2011-12-19
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Per t.b.'s comment, your space isn't $L^p[a,b]$. $L^p[a,b]$ is the space of measurable functions that are $p$-integrable over $[a,b]$ ( $\int_a^b |f|^p<\infty$).
That these spaces are complete is known, to some, as the Riesz-Fischer theorem. The standard proof of this theorem uses the fact that a normed space is a Banach Space if and only if every absolutely summable series is summable.
If you want a proof that $L_p$ is complete, without using the above fact ("using Cauchy sequences", as you ask), see here (this proof essentially incorporates that fact, though).
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0Thank you for your kind and quick response! – 2011-12-19