From what I gather, when working with PDEs we can show the uniqueness of a solution to certain PDE's by requiring that a bilinear form is continuous and coercive. The coercivity of the bilinear form implies injectivity. So why don't we require that that instead of being coercive, that the bilinear form is in fact, injective?
Also, coercivity implies injectivity, so surely limiting the bilinear form to being coercive rather than injective, means we miss out on injective bilinear forms that are injective but not coercive? So why limited is this ok...what is wrong with bilinear forms that are injective but not coercive?
The fact that the linear equation $Lu=f$ has uniqueness of solutions is obviously equivalent to injectivity of $L$. But how to check such injectivity in practice?
For finite-dimensional equations, one can use linear algebra: determinants, Gaussian elimination, and so on. None of these methods works in the infinite dimensional setting of PDEs. However, it turns out that the coercivity of the quadratic form $Q(f)=\langle Lf, f\rangle$ is a sufficient (not necessary) condition for injectivity. This is good, because coercivity of $Q$ is something that one can often check by integration by parts or other techniques.