---
tag: 9BS2PM0227
kind: definition
name: Coercive bilinear form
msc: [47A07, 35A15]
defines: 86KRHS6X22
uses: [0SQZMXB6GP, HM2HHE3J51]
sources:
  - url: https://en.wikipedia.org/wiki/Coercive_function
---

## Definition

Let $H$ be a real [[Hilbert space|HM2HHE3J51]] with norm $\norm{\cdot}$.
A [[bilinear form|0SQZMXB6GP]] $a \colon H \times H \to \R$ is *coercive* if there is a constant $\alpha > 0$ such that
$$a(u, u) \ge \alpha \norm{u}^2 \quad \text{for all } u \in H.$$

## Remarks

Example: the inner product $a(u, v) = \ip{u}{v}$ is coercive with $\alpha = 1$, since $a(u, u) = \norm{u}^2$.
Non-example: on $\R^2$, the form $a(u, v) = u_1 v_1$ is not coercive, because $a(u, u) = 0$ for $u = (0, 1)$, while $\norm{u}^2 = 1$.
The same definition is used on a real normed space.
On a complex Hilbert space, a sesquilinear form $a$ is called coercive if $\operatorname{Re} a(u, u) \ge \alpha \norm{u}^2$ for all $u$.
Some authors call a coercive form elliptic.
In the Lax–Milgram theorem the form is also assumed bounded: $\abs{a(u, v)} \le M \norm{u} \norm{v}$ for some constant $M$ and all $u, v \in H$.