Abstract
In this paper for real number , we define a function-valued inner product on by using dilation operator. We consider the relationship between orthonormal sequences, orthonormal bases and frames associated to the function-valued inner product and orthonormal sequences, orthonormal bases and frames associated to the usual inner product in . Also the Bessel inequality and the relation between bounded linear operators on associated to the function-valued inner product known function-valued factorable operators and frames respect to the function-valued inner product are discussed.
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