Let \((\Omega, \Sigma, \mu)\) be a meausurable space and \(\mu(\Omega)<\infty\). Then for \(p,q\in [1,\infty]\) with \(p>q\) there is a constant \(C>0\) which depends on \(\mu(\Omega)\), \(p\) and \(q\) such that

\begin{equation*} \lVert f\rVert_{L^q(\mu)}\le C\lVert f\rVert_{L^p(\mu)} \end{equation*}

for all \(f\in L^p(\mu)\).

That means, that \(L^p(\mu)\) is continuously embedded in \(L^q(\mu)\), i.e. \(L^p(\mu)\hookrightarrow L^q(\mu)\).

See also Link to heading

• The map \(p\mapsto \lVert x\rVert_p\) is monotonically decreasing for \(x\in\mathbb{K}^d\).