# Why the space have to be complex in the spectral theorem for bounded self-adjoint operators?

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?

• I guess for a real Hilbert space H, you can always complexify it to get a complex Hilbert space K and extend the your operator to be defined on K. After this extension, your operator should still be self-adjoint, then you can apply the spectral theory. – lye012 May 12 at 21:42
• math.stackexchange.com/questions/638216/… – DisintegratingByParts May 13 at 3:32