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\title{$L^p$ spaces in Isabelle}
\author{Sebastien Gouezel}
\date{}
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\begin{abstract}
$L^p$ is the space of functions whose $p$-th power is integrable. It is one
of the most fundamental Banach spaces that is used in analysis and
probability. We develop a framework for function spaces, and then implement
the $L^p$ spaces in this framework using the existing integration theory in
Isabelle/HOL. Our development contains most fundamental properties of $L^p$
spaces, notably the H\"older and Minkowski inequalities, completeness of
$L^p$, duality, stability under almost sure convergence, multiplication of
functions in $L^p$ and $L^q$, stability under conditional expectation.
\end{abstract}

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