Hellinger–Toeplitz theorem

The set of Hellinger - Toeplitz is a mathematical theorem from functional analysis. It is named after the mathematicians Ernst Hellinger and Otto Toeplitz.

Formulation

There are a Hilbert space and a symmetrical linear operator, that is, an operator that all the equation

Met. Then is continuous.

Evidence

After the sentence from the closed graph theorem it is sufficient to show the following: If a null sequence and convergent, then. Used to set the continuity of the scalar product, and then follows

Therefore.

Conclusions

• Since the operator is linear and continuous, it is also limited.
• Each symmetrical around defined operator is self-adjoint.
• Unlimited self-adjoint operators can be defined at most on a dense subset of a Hilbert space.

Generalization

One can weaken the condition in the theorem of Hellinger - Toeplitz:

Let and be Hilbert spaces and a linear operator, which has an adjoint, which means that there is an operator which for all and the equation

Met. Then and steadily.

The proof is analogous.