4 Function Spaces and Functional Analysis
4.1 Why function spaces enter engineering
An approximate solution may be continuous but not twice differentiable across element boundaries. That is acceptable if the variational problem is posed in a space where the required integrals make sense.
\(L^2(\Omega)\) is the space of square-integrable functions (modulo equivalence almost everywhere). \(H^1(\Omega)\) contains functions whose values and first weak derivatives are square integrable. The subspace \(H_0^1(\Omega)\) represents zero trace on the appropriate boundary for standard homogeneous Dirichlet conditions.
4.2 Norms and bilinear forms
For a scalar function \(v\), a common \(H^1\) norm is
\[\lVert v\rVert_{H^1}^2=\lVert v\rVert_{L^2}^2+\lVert\nabla v\rVert_{L^2}^2.\]
A weak problem often asks for \(u\in V\) such that \(a(u,v)=\ell(v)\) for all \(v\in V\), where \(a\) is a bilinear form and \(\ell\) is a continuous linear functional. Existence, uniqueness, and stability require mathematical conditions appropriate to the problem.
4.3 Why this helps with FEM
Finite-element spaces are finite-dimensional subspaces of function spaces. Approximation quality and stability are then questions about the chosen spaces, the mesh, and the bilinear forms—not only about matrix dimensions.
Exercise: Explain why continuous piecewise-linear functions naturally belong to \(H^1\) on an interval but generally not to \(H^2\).
Living chapter. This is an initial working treatment. Derivations, examples, and references will be expanded in future editions.