A New Semi-Inner Product and pn-Angle in the Space of p-Summable Sequences

In this paper, we propose a definition for a semi-inner product in the space of p-summable sequences equipped with an n-norm. Using this definition, we introduce the concepts of pn-orthogonality and the pn-angle between two vectors in the space of p-summable sequences. For the special case n = 1, these concepts are identical to previous studies. We also introduce the notion of the pn-angle between one-dimensional subspaces and arbitrary-dimensional subspaces. The authors believe that the results obtained in this paper are very significant, especially in the theory of n-normed space in functional analysis.