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Spaces $L^p$ of $\Omega$

The spaces $L^p(\Omega)$ are functional spaces that describe integrable functions on a set $\Omega$. Specifically, a function $f$ is in the space $L^p(\Omega)$ if its $L^p$ norm is finite, where the norm is defined as:

$$\|f\|_{L^p(\Omega)} = \left( \int_{\Omega} |f(x)|^p \mathrm{d}x \right)^{1/p}$$
\[\|f\|_{L^p(\Omega)} = \left( \int_{\Omega} |f(x)|^p \mathrm{d}x \right)^{1/p}\]

where $\lvert f(x) \rvert^p$ represents the absolute value of $f(x)$ raised to the power $p$, and $\mathrm{d}x$ represents the volume element in the space $\Omega$. The spaces $L^p(\Omega)$ are Banach spaces for $1 \leq p < \infty$.

Definition of the space $L^1(\Omega)$

The space $L^1(\Omega)$ is the space of functions $f$ such that:

$$\int_{\Omega} |f(x)| \mathrm{d}x < \infty$$
\[\int_{\Omega} |f(x)| \mathrm{d}x < \infty\]

Definition of the space $L^2(\Omega)$

The space $L^2(\Omega)$ is the space of functions $f$ such that:

$$\int_{\Omega} |f(x)|^2 \mathrm{d}x < \infty$$
\[\int_{\Omega} |f(x)|^2 \mathrm{d}x < \infty\]

Definition of the $L^\infty(\Omega)$ space

The space $L^\infty(\Omega)$ consists of functions $f$ such that:

$$\|f\|_{L^\infty(\Omega)} = \inf \{M \in \mathbb{R} : |f(x)| \leq M \text{ a.e. on } \Omega\}$$
\[\|f\|_{L^\infty(\Omega)} = \inf \{M \in \mathbb{R} : |f(x)| \leq M \text{ a.e. on } \Omega\}\]

where “a.e.” means “almost everywhere”. In other words, the norm $\lVert f\rVert_{L^\infty(\Omega)}$ is the smallest value $M$ such that $\lvert f(x)\rvert \leq M$ holds almost everywhere on $\Omega$. Functions in $L^\infty(\Omega)$ are sometimes called “essentially bounded” or “essentially supremum-bounded” functions.

Examples of using the $L^p$ norm

When $p = 2$, the $L^p$ norm corresponds to the Euclidean norm in a Hilbert space, and is often denoted $|f|_{2}$. For a function $f \in L^2(\Omega)$, we have:

$$\|f\|_2 = \sqrt{\int_{\Omega} |f(x)|^2 \mathrm{d}x}$$
\[\|f\|_2 = \sqrt{\int_{\Omega} |f(x)|^2 \mathrm{d}x}\]

The Hölder inequalities

The Hölder inequalities are inequalities that allow one to bound an integral of a product of functions in an $L^p(\Omega)$ space in terms of the $L^p$ norms of the individual functions. More precisely, if $p$ and $q$ are exponents such that $1/p + 1/q = 1$, then for any functions $f$ and $g$ in $L^p(\Omega)$, we have:

$$\left|\int_{\Omega} f(x) g(x) \mathrm{d}x \right| \leq \|f\|_{L^p(\Omega)} \|g\|_{L^q(\Omega)}$$
\[\left|\int_{\Omega} f(x) g(x) \mathrm{d}x \right| \leq \|f\|_{L^p(\Omega)} \|g\|_{L^q(\Omega)}\]

These inequalities have important applications in functional analysis, probability theory, and mathematical physics.