Let \(A_ n\) (respectively \(H^ 1_ n)\) be a finite dimensional space, generated by the functions \(z^ k\), \(k=0,1,...,n-1\) and provided by the norm \(\| f\|_ A=\max_{| z| =1}| f(z)|\) (respectively, \[ \| f\|_{H^ 1}=\sup_{0<r<1}(1/\pi)\int^{2\pi}_{0}| f(re^{i\theta})| d\theta). \] The existence in spaces \(A_ n\) (respectively, \(H^ 1_ n)\) of bases (respectively, of unconditional bases), whose basis constant does not depend on the dimension n, is proved.