Recently, I was studying the spectrum of an operator T on a normed
linear space X. In order to ensure that the spectrum is non-empty, one
needs to assume that X is a complex Banach space. But the books don't give
sufficient justification. I would like to have an example of a real
incomplete normed linear space as well as an example of a complex
incomplete normed linear space X and a bounded operator T: X ----> x such
that the spectrum of T is empty. In order to avoid confusion, I would like
to define the spectrum: A scalar $\lambda$ is called a spectral value of a
bounded operator T on a normed linear space X if T - $\lambda$I is not
invertible. 'I' is the identity operator on X and the set of all spectral
values of T is called the spectrum of T.