Tikhonov's theorem
Encyclopedia
Tikhonov's theorem or Tychonoff's theorem can refer to any of several mathematical theorems named after the Russian mathematician Andrey Nikolayevich Tikhonov:
- Tychonoff's theoremTychonoff's theoremIn mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact. The theorem is named after Andrey Nikolayevich Tychonoff, who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the...
, which states that the productProduct topologyIn topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology...
of any collection of compactCompact spaceIn mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
topological spaceTopological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s is compact - Tikhonov's fixed point theorem, concerning fixed points of continuous mappings on compact subsets of certain topological vector spaces
- Tikhonov's theorem (dynamical systems)Tikhonov's theorem (dynamical systems)In applied mathematics, Tikhonov's theorem on dynamical systems is a result on stability of solutions of systems of differential equations. It has applications to chemical kinetics...
, concerning limiting behaviour of solutions of systems of differential equations applicable to (among other things) chemistry - Tychonoff's uniqueness theorem, concerning the one-dimensional heat equationHeat equationThe heat equation is an important partial differential equation which describes the distribution of heat in a given region over time...