- joint continuity: for each neighbourhood of zero there are neighbourhoods of zero and such that (in other words, this condition means that the multiplication is continuous as a map between topological spaces ), or
- stereotype continuity: for each totally bounded set and for each neighbourhood of zero there is a neighbourhood of zero such that and , or
- separate continuity: for each element and for each neighbourhood of zero there is a neighbourhood of zero such that and .
(Certainly, joint continuity implies stereotype continuity, and stereotype continuity implies separate continuity.) In the first case is called a "topological algebra with jointly continuous multiplication", and in the last, "with separately continuous multiplication".
- 1. Fréchet algebras are examples of associative topological algebras with jointly continuous multiplication.
- 2. Banach algebras are special cases of Fréchet algebras.
- 3. Stereotype algebras are examples of associative topological algebras with stereotype continuous multiplication.
- Beckenstein, E.; Narici, L.; Suﬀel, C. (1977). Topological Algebras. Amsterdam: North Holland. ISBN 9780080871356.
- Akbarov, S.S. (2003). "Pontryagin duality in the theory of topological vector spaces and in topological algebra". Journal of Mathematical Sciences. 113 (2): 179–349. doi:10.1023/A:1020929201133. S2CID 115297067.
- Mallios, A. (1986). Topological Algebras. Amsterdam: North Holland. ISBN 9780080872353.
- Balachandran, V.K. (2000). Topological Algebras. Amsterdam: North Holland. ISBN 9780080543086.
- Fragoulopoulou, M. (2005). Topological Algebras with Involution. Amsterdam: North Holland. ISBN 9780444520258.