For $\Omega=(0,1). $Prove that there exists $M>0$ such that $$||u||_{C^0(\overline{\Omega})}\le M||u||_{H^1(\Omega)}$$ for all $u\in H^1(\Omega).$
2026-04-23 01:55:00.1776909300
Inequality for embedding in Sobolev space
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Assume that $u(0)=0$. Then $$ u(x)=\int_0^x u'(s)ds=\int_0^1 \textbf{1}_{(0,x)}u'(s)ds. $$ Now you use Cauchy-Schwarz inequality. $$ u(x)\leq \|u'\textbf{1}_{(0,x)}\|_{L^2}=\int_0^1 (u'(s))^2\textbf{1}_{(0,x)}ds\leq \int_0^1 (u'(s))^2 ds. $$ Right?