  In this paper, we consider real and complex algebras as well as algebras over
general fields. In Section 2, we revisit and prove several results on
(quadratic) algebras over general fields. In Section 3, we show how to develop
the spectral theory in the context of complex (resp. real) one-sided
alternative topological algebras. As an application of the existence of
spectrum, we prove the existence of nontrivial hyperinvariant linear manifolds
for nonscalar (resp. nonquadratic) continuous linear operators acting on
complex (resp. real) Fr\'echet spaces. Along the way, spectral theory is used
to prove several topological counterparts of the well-known theorems of
Frobenius, Hurwitz, Gelfand-Mazur, and Zorn. This is done, for example, in the
context of left (resp. right) alternative topological algebras whose duals
separate their elements. In Section 4, we consider real and complex algebras in
various topological settings and reconsider and prove several results. For
instance, it is shown that given a $ 1 < k \in \mathbb{N}$, on any locally
complex algebra, there exists a unique nonzero vector space norm, say, $\|.\|$,
satisfying the identity $\|a^k\| = \|a\|^k$ on the algebra. In Section 5, among
other things, we revisit and slightly strengthen the celebrated theorems of
Frobenius, Zorn, Gelfand-Mazur, and Hurwitz, and also give slight extensions of
their topological counterparts, e.g., theorems of Albert, Kaplansky, and
Urbanik-Wright to name a few, in several settings.
