Let $\eH$  be a complex Hilbert space and let $\pS_{\eH}=\{ x\in \eH;~\| x\|=1\}$ be the unit sphere of $\eH$. Every bounded linear operator $A$ on $\eH$ defines a quadratic form as follows

\begin{equation*}

q_A:\quad \left. \begin{array}{llc} \pS_{\eH}& \longrightarrow & \bC\\[1mm]

x& \mapsto &\langle Ax,x\rangle, \end{array} \right.

\end{equation*}

where $\langle \cdot,\cdot \rangle$ denotes the inner product of $\eH$.

The image of $q_A$ is the numerical range $W(A)\subset \bC$ of $A$. It is not hard to see that the spectrum of an operator is a subset of the closure of the numerical range, which means that numerical ranges are a useful tool in locating the spectrum. Some classical results about numerical ranges will be presented; for instance, the Toeplitz-Hausdorff Theorem and the Hildebrandt's Theorem.

The hyperreflexivity of sets of operators determined by the numerical range will be discussed, as well.