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Let $p\geq1$ be a real number and $L_p([a,b])=L_p([a,b],\mathcal{B}, \lambda)$ be the normed space which consists of all equivalence classes of $\mathcal{B}$-measurable functions $f$ on $[a,b]$ to $\mathbb R$ such that $\int_{[a,b]}|f|^p d\lambda<\infty$. Here,

  • $\mathcal B$ is the Borel algebra,
  • $\lambda$ is the Lebesgue measure,
  • $f\sim g \Leftrightarrow f=g$ a.e. and
  • the norm $||\cdot||_p$ is given by $$||f||_p=\left(\int_{[a,b]}|f|^p d\lambda\right)^{1/p}.$$

Moreover, let $C[a,b]$ be the set of continuous functions on $[a,b]$ to $\mathbb R$. Then, make $M:=\left\{\overline{f}:f\in C[a,b]\right\}$. I'd like to show that the completion of the normed space $(M, ||\cdot||_p)$ is $(L_p([a,b]),||\cdot||_p)$. I guess all I need to do is to prove that $M$ is dense in $(L_p([a,b])$.

I could find the proof of more general statements, for example, in Reddy's Introductory Functional Analysis or in Nair's Functional Analysis: a first course. But, for simplicity, I'd like to discover a reference with deals specifically with this particular case. Could someone indicate a book with this proof? Thanks in advance!

rgm
  • 817

1 Answers1

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Hint. Proving that $L_p([a,b])$ is the completion of $M$ is to show that there is a subspace $W$ of $L_p([a,b])$ such that $W$ is dense in $L_p([a,b])$ and $W$ is isometric with $M$ in the sense that there is a bijective isometry from $M$ to $W$. Since any inclusion is a bijective isometry, it's suffices to verify that $M$ is dense in $L_p([a,b])$. This verification can be found on this post. There, were used Lusin's theorem, Tietze extension theorem and a proposition of Royden and Fitzpatrick's Real Analysis.

rgm
  • 817