In an exercise, i would like to use the next afirmation.
Si $L=\delta_0$ y $v\in H^1_0$ entonces
$|<L,v>| =|v(0)| \le ||v||_\infty \le K_1 ||v||_{H^1} \le K_2 ||v||_{H^1_0}.$
But, i do not know if Its true that $||v||_\infty \le K_1 ||v||_{H^1}$. Is it a consequence of v is, in particular, continous ($H^1_0\subset C^0)$? All of this in a domain $\Omega$ one dimensional.
Yes, it is true that $H_0^1(\Omega) \subset C^0(\Omega)$ when $\Omega \subset \mathbb{R}$. This follows from the (second part of the) Sobolev embedding theorem as written here. In fact, listed there is the stronger result that $$H_0^1(\Omega) \subseteq C^{0,\frac12}(\Omega)$$ (the space of $1/2$-Holder continuous functions) with the embedding being continuous which gives the desired inequality.