In the spectral theorem for compact self-adjoint linear operators $T:H\to H$ (as stated in Conway's book), the Hilbert space $H$ can be real or complex.
However, in the spectral theorem for bounded self-adjoint linear operators $T:H\to H$ (as stated in Bachman's book), the Hilbert space $H$ have to be complex.
Is this assumption really needed? Is there any version for bounded self-adjoint linear operators defined on real spaces?