Lax-Milgram Theorem

Let $\phi$ be a bounded Coercive bilinear Functional on a Hilbert Space $H$. Then for every bounded linear Functional $f$ on $H$, there exists a unique $x_f\in H$ such that

$$f(x)=\phi(x, x_f)$$

for all $x\in H$.

References

Debnath, L. and Mikusinski, P. Introduction to Hilbert Spaces with Applications. San Diego, CA: Academic Press, 1990.

Zeidler, E. Applied Functional Analysis: Applications to Mathematical Physics. New York: Springer-Verlag, 1995.