Riesz Representation Theorem

The properties of the linear functionals in Hilbert space are similar to the properties of the linear functionals in Euclidean space. For example, consider a fixed element 1 of a Hilbert space H. Then we can introduce a linear functional  

We have demonstrated in Appendix A that any linear functional in Euclidean space can be represented in a form of (C.2). We can prove now that the same fact is correct in the general case of the functionals in Hilbert space. 

Every bounded linear functional /(x) in a Hilbert space may be represented as (1, x) and 1 is uniquely determined by /. 

Consider a basis ei, e2, eg,.e ,. of the Hilbert space. We know that for 

we have proved that the vector 1 is uniquely determined by /. 

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According to the Riesz representation theorem there exist vectors (elements of the space M) such that... 

As an example of practical application of the Riesz representation theorem, consider the 2-D gravity problem. The forward gravity operator is given by the formula... 

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