The Proof Of The Several Equivalent Definitions Of Riesz Base With Spectral Measure

In this paper,we will prove that the following definitions are equivalent with spectral measure in the Hilbert space,(1){xn}n=1∞is a Riesz basis for H;(2)span{xn}=H,there exist constants A,B with0<4≤B<∞such that for every p∈N and complex numbers cn,we have(3)span{xn}=H,T={<xn,xm>}n,m=1∞:l2(N)â†'l2(N)is a bound-ed,invertible operator on l2(N);(4)span{xn}=H,{xn} have a complete biorthogonal sequence{gn} such that for every f∈H...