An approach to F. Riesz representation Theorem

The Riesz representation theorem

The theorem of F. and M. Riesz

Suppose that G is a locally compact abelian group and that P ⊂ G is a semigroup (satisfies P + P ⊂ P ) that is closed and satisfies P ∩ (−P ) = {0} and P ∪ (−P ) = G. We define a total order on G by x ≤ y when y − x ∈ P . We verify that this is indeed a total order. (We remark that nowhere in this do we show the significance of P being closed; but in this note we shall be speaking about discret...

A New Approach to Continuous Riesz Bases

This paper deals with continuous frames and continuous Riesz bases. We introduce continuous Riesz bases and give some equivalent conditions for a continuous frame to be a continuous Riesz basis. It is certainly possible for a continuous frame to have only one dual. Such a continuous frame is called a Riesz-type frame [13]. We show that a continuous frame is Riesz-type if and only if it is a con...

A Formal Proof Of The Riesz Representation Theorem

This paper presents a formal proof of the Riesz representation theorem in the PVS theorem prover. The Riemann Stieltjes integral was defined in PVS, and the theorem relies on this integral. In order to prove the Riesz representation theorem, it was necessary to prove that continuous functions on a closed interval are Riemann Stieltjes integrable with respect to any function of bounded variation...

The Orthogonal Projection and the Riesz Representation Theorem

In this article, the orthogonal projection and the Riesz representation theorem are mainly formalized. In the first section, we defined the norm of elements on real Hilbert spaces, and defined Mizar functor RUSp2RNSp, real normed spaces as real Hilbert spaces. By this definition, we regarded sequences of real Hilbert spaces as sequences of real normed spaces, and proved some properties of real ...