1. **Problem Statement:** We want to understand the ideal of a linear function in a Hilbert space, focusing on the numerical range and maximal ideals. 2. **Background:** In a Hilbert space $H$, a linear operator $T: H \to H$ has a numerical range defined as $$ W(T) = \{ \langle Tx, x \rangle : x \in H, \|x\|=1 \} $$ where $\langle \cdot, \cdot \rangle$ is the inner product. 3. **Maximal Ideals:** In the algebra of bounded linear operators $\mathcal{B}(H)$, maximal ideals correspond to kernels of irreducible representations or characters. For commutative $C^*$-algebras, maximal ideals correspond to points in the spectrum. 4. **Relation to Numerical Range:** The numerical range $W(T)$ is a convex subset of the complex plane containing the spectrum $\sigma(T)$. Maximal ideals relate to spectral properties, and the numerical range gives insight into operator behavior. 5. **Summary:** The ideal generated by a linear function/operator in a Hilbert space is closely linked to its spectral properties, with the numerical range providing a convex set containing the spectrum, and maximal ideals corresponding to points in the spectrum or irreducible representations. This explanation provides a conceptual understanding rather than a single formula, as the problem is abstract and theoretical.