# Function Spaces¶

To use the boundary element method, we start with a variational boundary integral equation, for example: Find $$u\in H^{1/2}(\Gamma)$$ such that for all $$v\in H^{1/2}(\Gamma)$$,

$\left\langle\mathsf{V}u,v\right\rangle = \left\langle f,v\right\rangle.$

An approximation of the solution of this problem is then found by discretising the problem and searching for a solution in a subspace $$\mathcal{V}_h\subset H^{1/2}(\Gamma)$$.

In this section of the Bempp Handbook, we look at the definitions of continuous function spaces such as $$H^{1/2}(\Gamma)$$ that are used in the boundary integral equations, and we look at how discrete subspaces of these are usually defined.