Functional

In Hilbert spaces
Let $$\mathcal H$$ be a Hilbert space. A linear operator $$\Phi:\mathcal H \to \mathbb C$$ is called a functional (Def 4.4.1 ).

An interesting property of functionals is that, any bounded functional can be defined as

$$\Phi \mathbf v :=\langle\mathbf{v,w}\rangle$$ for some $$\mathbf w$$