Recently, I was studying the spectrum of an operator T$T$ on a normed linear space X$X$. In order to ensure that the spectrum is non-empty, one needs to assume that X$X$ 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$X$ and a bounded operator T: X ----> x$T:X \rightarrow X$ such that the spectrum of T$T$ 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$T$ on a normed linear space X if T -$X$ if $\lambda$I$T - \lambda I$ is not invertible. 'I''$I$' is the identity operator on X$X$ and the set of all spectral values of T$T$ is called the spectrum of T$T$.
