\input zb-basic
\input zb-matheduc
\iteman{ZMATH 2007f.00384}
\itemau{Kulkarni, S.H.}
\itemti{A very simple and elementary proof of a theorem of Ingelstam.}
\itemso{Am. Math. Mon. 111, No. 1, 54-58 (2004).}
\itemab
The theorem in question states that a unital Banach algebra over the field $\Bbb R$ whose underlying Banach space is a (real) Hilbert space is isomorphic to one of the algebras $\Bbb R$, $\Bbb C$ or $\Bbb H$ (=~quaternions), cf.\ [{\it I.~Ingelstam}, Math. Scand. 11, 22--32 (1962; Zbl 0122.35003)]. The paper under review certainly lives up to its title since no Banach algebra theory and barely any Hilbert space theory is used; incidentally, completeness turns out to be inessential for the theorem. The only criticism I have is that the author's reasoning for Fact~1 only works if the set $S$ is assumed orthonormal and that the name Schwarz in Cauchy-Schwarz inequality is misspelt two thirds of the time.
\itemrv{Dirk Werner (Berlin)}
\itemcc{I95}
\itemut{real Banach algebra; Hilbert space}
\itemli{doi:10.2307/4145018}
\end