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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?2011-12-19
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    Your 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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    Furthermore, 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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    In that case it is complete by definition.2011-12-19

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