let $V_{C} = \{u \in W^{1,2}(0,1)\,\, | \,\, u(1) = Cu(0)\}, \,\, C \in \mathbb{R}$

  1. show that $V_C$ is a closed subspace of $W^{1,2}(0,1)$

I got stuck trying to solve this here's my work : if we let $T : W^{1,2}(0,1) \to \mathbb{R}$ such that $T(u) = u(1)-Cu(0)$

then $V_C = \ker T$ and $T(u) = C(u(1) - u(0)) + (1-C)u(1) \implies |T(u)| \leq ||u'||_{L^1(0,1)} + K||u||_{\infty} $

we're not in a finite dimension space so I can't move forward

any help will be appreciated.

  • $\begingroup$ Do you know the estimates $\|u'\|_{L^1} \le \|u'\|_{L^2}$ and $\|u\|_\infty \le C \|u\|_{W^{1,2}}$? $\endgroup$ – gerw Jun 20 at 10:35
  • $\begingroup$ @gerw the second one is new to me but it solves my problem, thanks, I'll look it up, does it have a particular name ? $\endgroup$ – rapidracim Jun 20 at 12:23
  • 1
    $\begingroup$ It is a particular case of Sobolev's embedding theorem. $\endgroup$ – gerw Jun 20 at 18:07

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