Final answer:
Every bounded linear functional on a Hilbert space is represented by a unique element in the space, as stated by the Riesz Representation Theorem, implying an isometric correspondence between the bounded linear functionals and the Hilbert space itself.
Step-by-step explanation:
The characterization of all bounded linear functionals on a Hilbert space can be understood through the Riesz Representation Theorem.
According to the theorem, for every bounded linear functional f on a Hilbert space H, there exists a unique element y in H such that f(x) = < x, y > for all x in H. Here, < x, y > denotes the inner product on the Hilbert space. This correspondence between bounded linear functionals and elements of the Hilbert space is isometric, showing that the dual of a Hilbert space is isomorphic to the space itself.
Bounded linear functionals are important in various applications because they are continuous and adhere to a norm bound: | f(x) | ≤ C | x | for some constant C and all x in H. Such functionals are crucial in functional analysis and quantum mechanics, where they often represent observable quantities.