= Functional inequality
A functional inequality bounds quantities associated with an entire class of <functions>, such as their norms, integrals or derivatives. Its hypotheses, domain, boundary conditions and constant are part of the statement. <Poincaré inequalities> and <Sobolev inequalities> are examples that control a function using derivative information.
For an absolutely continuous real function $u$ on $[0,1]$ with $u(0)=0$ and $u'\in L^2$, the <Cauchy-Schwarz inequality> gives
$$
|u(x)|^2=\left|\int_0^xu'(t)dt\right|^2\leq x\int_0^1|u'(t)|^2dt.
$$
Integrating in $x$ yields $\int_0^1|u|^2\leq\frac12\int_0^1|u'|^2$. This constant is sufficient for the argument; no optimality is asserted.
Without a boundary or normalization condition the derivative cannot control a nonzero constant function, whose derivative vanishes. Such inequalities explain how energy estimates can control solutions of <partial differential equations> and give coercive bounds in <variational problems>.
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