Can any infinite dimensional Hilbert space have countably infinite dimension?
I know that there doesn't exist any infinite dimensional Banach space whose dimension is countably infinite for otherwise it would violate Baire's category theorem. What about infinite dimensional Hilbert spaces? Since Hilbert spaces are also Banach spaces the same result has to be true for Hilbert spaces. Isn't it so? Am I missing something?
Any suggestion regarding this will be highly appreciated. Thanks in advance.