Boundary Distance Functions of Riemann Domains Over Pre-Hilbert Spaces

In the present paper, we generalize subpluriharmonic functions in the sense of Fujita to infinite dimension. We show that, for every open set D in a complex normed space E and for the boundary distance function d of D , the function - ln d is subpluriharmonic on D . Moreover, we show that, for every Riemann domain ( D , π ) over a complex pre-Hilbert space E and for the boundary distance function d of ( D , π ) , the function - ln d is locally subpluriharmonic on D .