1. The problem is to understand the concept of an ideal of a linear function in a Hilbert space, focusing on the numerical range and maximal ideal. 2. In a Hilbert space $\mathcal{H}$, a linear operator $T: \mathcal{H} \to \mathcal{H}$ has a numerical range defined as $$ W(T) = \{ \langle Tx, x \rangle : x \in \mathcal{H}, \|x\|=1 \} $$ where $\langle \cdot, \cdot \rangle$ is the inner product. 3. The numerical range $W(T)$ is a subset of the complex plane and is convex by the Toeplitz-Hausdorff theorem. 4. An ideal in the algebra of bounded linear operators on $\mathcal{H}$ is a subset closed under addition, scalar multiplication, and multiplication by any operator in the algebra. 5. A maximal ideal is an ideal that is maximal with respect to inclusion, meaning it is not contained in any larger proper ideal. 6. The study of numerical ranges helps characterize spectral properties of operators and relates to maximal ideals by identifying operators whose numerical range lies in certain subsets of the complex plane. 7. Understanding these concepts is crucial in operator theory and functional analysis, providing insight into the structure of operator algebras and their ideals. Final summary: The ideal of a linear function in a Hilbert space is closely linked to its numerical range, which is convex and provides spectral information. Maximal ideals represent the largest proper subsets closed under operator multiplication, and analyzing numerical ranges aids in identifying and understanding these maximal ideals.