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 $T:X \rightarrow X$ 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 $T$.