# closed subspace of $W^{1,2}(0,1)$

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.

• Do you know the estimates $\|u'\|_{L^1} \le \|u'\|_{L^2}$ and $\|u\|_\infty \le C \|u\|_{W^{1,2}}$? – gerw Jun 20 at 10:35
• @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 ? – rapidracim Jun 20 at 12:23
• It is a particular case of Sobolev's embedding theorem. – gerw Jun 20 at 18:07