a bilinear form
When you rewrite a PDE in its weak form, the left-hand side becomes a machine that eats two functions — the unknown u and a test function v — and returns a number, while being linear in each slot separately. That machine is a bilinear form. It is the exact infinite-dimensional analogue of the dot product or of the expression v-transpose A u from linear algebra, and recognizing it as such is what lets the whole linear-algebra toolkit transfer to PDE.
Formally, a bilinear form a(u, v) on a function space takes two inputs and is linear in u when v is held fixed, and linear in v when u is held fixed: a(c1 u1 + c2 u2, v) = c1 a(u1, v) + c2 a(u2, v), and likewise in the second slot. The one that arises from minus Laplacian u = f is a(u, v) = integral of grad u . grad v dx — manifestly linear in each of u and v. More general second-order elliptic operators give a(u, v) = integral of (A grad u . grad v + b . grad u v + c u v), still bilinear. Two properties of this form decide everything downstream. It is bounded (continuous) if |a(u, v)| is at most M times ||u|| ||v|| — the machine does not amplify size without limit. It is coercive if a(u, u) is at least alpha times ||u||^2 — feeding the same function into both slots always yields something positive and proportional to the function's size. These are the two hypotheses of the Lax-Milgram theorem.
The reason this abstraction pays off: once a problem reads a(u, v) = L(v) for all v, with a bounded and coercive and L a bounded linear functional, Lax-Milgram instantly grants a unique solution u — no matter how complicated the original PDE looked. The bilinear form also carries the geometry: when a is symmetric, a(u, v) = a(v, u), it is literally a new inner product, the energy inner product, and solving a(u, v) = L(v) is the same as minimizing the energy (1/2) a(u, u) - L(u) (Dirichlet's principle). Symmetry of the form is exactly the fingerprint of an equation that comes from an energy; a non-symmetric form (one with a drift term b . grad u) signals an equation with no underlying energy, where you must use Lax-Milgram directly rather than minimization.
For the Laplacian, a(u, v) = integral of grad u . grad v. Boundedness: by Cauchy-Schwarz, |a(u, v)| is at most ||grad u||_{L^2} ||grad v||_{L^2}, at most ||u||_{H^1} ||v||_{H^1}. Coercivity on H^1_0: a(u, u) = integral of |grad u|^2, and by Poincare this is at least a constant times ||u||_{H^1}^2. Both Lax-Milgram hypotheses hold, so minus Laplacian u = f has a unique weak solution.
Boundedness and coercivity of a(u,v) are the two boxes you tick to invoke Lax-Milgram.
A bilinear form need not be symmetric — drift (convection) terms b . grad u make it non-symmetric, and then it is NOT an inner product and there is no energy to minimize, yet Lax-Milgram still applies because it never required symmetry. Conversely, coercivity is not automatic: an equation that is not coercive (an indefinite or sign-changing form) can fail to have a unique solution, which is precisely the regime where the Fredholm alternative takes over.