Banach Algebra

A Banach algebra is an Algebra $B$ over a Field $F$ endowed with a Norm $\Vert\cdot\Vert$ such that $B$ is a Banach Space under the norm $\Vert\cdot\Vert$ and multiplication is continuous in the sense that if $x,y\in B$ then $\Vert xy\Vert\leq \Vert x\Vert\,\Vert y\Vert$. Continuity of multiplication is the most important property.

$F$ is frequently taken to be the Complex Numbers in order to assure that the Spectrum fully characterizes an Operator (i.e., the spectral theorems for normal or compact normal operators do not, in general, hold in the Spectrum over the Real Numbers).

See also B*-Algebra