---
tag: 86KRHS6X22
kind: concept
name: Coercive bilinear form
msc: [47A07, 35A15]
specializes: [0SQZMXB6GP]
sources:
  - url: https://en.wikipedia.org/wiki/Coercive_function
---

## Summary

A coercive bilinear form on a real Hilbert space is a [[bilinear form|0SQZMXB6GP]] $a$ with $a(u, u) \ge \alpha \norm{u}^2$ for a fixed $\alpha > 0$.
Coercivity is the key hypothesis of the Lax–Milgram theorem.
It turns the weak form of many elliptic partial differential equations into a problem with exactly one solution.

## Explanations

### L1 human

A bilinear form takes two vectors and returns a number, like the dot product.
Coercive means that pairing a vector with itself always gives a value at least as large as a fixed fraction of its squared length.
So no nonzero vector is "invisible" to the form, and long vectors give large values.
For example, $a(u, v) = 2 u_1 v_1 + 3 u_2 v_2$ on $\R^2$ is coercive, since $a(u, u) \ge 2 (u_1^2 + u_2^2)$.