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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? $$\left\{x(t)\in C[a,b] : \int_a^b |x(t)|^p \;dt < \infty \right\};$$ $$||x(t)||=\left[\int_a^b|x(t)|^p \;dt\right]^\frac{1}{p}$$

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Is it a homework problem? What did you try? –  Davide Giraudo Dec 19 '11 at 12:56
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Your description of $L^p[a,b]$ is inaccurate. The space $L^p[a,b]$ of $p$-integrable functionsis 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 –  t.b. Dec 19 '11 at 13:08
    
Furthermore, your notation of the norm of $x$ is ugly. Maybe you forgot to take the closure of your space. –  Jonas Teuwen Dec 19 '11 at 13:31
    
In that case it is complete by definition. –  Jonas Teuwen Dec 19 '11 at 13:37
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2 Answers

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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thank you for your response. –  neli Dec 19 '11 at 14:06
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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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Thank you for your kind and quick response! –  neli Dec 19 '11 at 14:05
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