Help me to understanding the Lax-Milgram Theorem.
I've shown that there exits a bilinear form $$ a : H_0^1([0,1]) \times H_0^1([0,1]) \to \mathbb{R}, (u,v) \mapsto \int_0^1 ( u'v' - K uv), \quad \text{ where } K \text{ is a constant} $$ Does this imply by the Lax-Milgram theorem that the differential equation $y'' - Ky = 0$ with boundary conditions $y(0) = y(1) = 0$ has a unique solution?
I know that by the Lax-Milgram theorem there exists a unique $u \in H_0^1([0,1])$ such that for all bounded $f \in H_0^1([0,1])^*$ (linear), we have $$ a(u,v) = f(v). $$ Do I have to choose the constant zero function here to conclude?