Let \(1\le p< q\le\infty\).
If \((X,\mcm,\mu)\) is a measure space with \(\mu(X)<\infty\), then \(L^q(X)\subseteq L^p(X)\), and \(\norm{f}_p\le\mu(X)^{\frac1p-\frac1q}\norm{f}_q\) for \(f\in L^q(X)\), with \(\frac1\infty\coloneqq0\). In particular \(L^\infty(X) \subseteq L^2(X) \subseteq L^1(X)\).
For counting measure on \(\ZZ\), \(\ell^p(\ZZ)\subseteq\ell^q(\ZZ)\), and \(\norm{a}_q\le\norm{a}_p\) for \(a\in\ell^p(\ZZ)\). In particular \(\ell^1(\ZZ) \subseteq \ell^2(\ZZ) \subseteq \ell^\infty(\ZZ)\).
For Lebesgue measure on \(\RR^n\), neither \(L^p(\RR^n)\subseteq L^q(\RR^n)\) nor \(L^q(\RR^n)\subseteq L^p(\RR^n)\) holds.
For \(q=\infty\), \(\int_X\abs{f}^p\le\mu(X)\norm{f}_\infty^p\). For \(q<\infty\), apply Hölder’s inequality with exponents \(r=q/p\) and \(r'=q/(q-p)\) to \(\abs{f}^p\cdot1\): \[ \int_X\abs{f}^p\dmu\le\qty{\int_X\abs{f}^q\dmu}^{p/q}\mu(X)^{1-p/q}, \] and take \(p\)-th roots.
If \(\norm{a}_p=1\), then \(\abs{a_k}\le1\) for all \(k\), so \(\norm{a}_\infty\le1\) and, for \(q<\infty\), \(\sum_k\abs{a_k}^q\le\sum_k\abs{a_k}^p=1\). The general case follows by scaling \(a\) by \(1/\norm{a}_p\).
Let \(f_a(x)\coloneqq\abs{x}^{-a}\one_{\abs{x}\le1}\) and \(g_a(x)\coloneqq\abs{x}^{-a}\one_{\abs{x}>1}\). In polar coordinates, \(f_a\in L^r(\RR^n)\) if and only if \(ar<n\) (for \(r<\infty\)), and \(g_a\in L^r(\RR^n)\) if and only if \(ar>n\). If \(q<\infty\), choose \(a\) with \(n/q\le a<n/p\): then \(f_a\in L^p\setminus L^q\), and choosing \(a\) with \(n/q<a\le n/p\) gives \(g_a\in L^q\setminus L^p\). If \(q=\infty\), then \(f_a\in L^p\setminus L^\infty\) for \(0<a<n/p\), and the constant function \(1\) lies in \(L^\infty\setminus L^p\).