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Let , be two closed subsets (or two open subsets) of a topological space such that =, and let also be a topological space. If f : A → B {\displaystyle f:A\to B} is continuous when restricted to both X {\displaystyle X} and Y , {\displaystyle Y,} then f {\displaystyle f} is continuous.
Example.The unitary symplectic group or quaternionic unitary group, denoted Sp(N) or U(N, H), is the group of all transformations of H N which commute with right multiplication by the quaternions H and preserve the H-valued hermitian inner product
A space is an absolute neighborhood retract for the class , written (), if is in and whenever is a closed subset of a space in , is a neighborhood retract of . Various classes C {\displaystyle {\mathcal {C}}} such as normal spaces have been considered in this definition, but the class M {\displaystyle {\mathcal {M}}} of metrizable spaces ...
In mathematics, the adele ring of a global field (also adelic ring, ring of adeles or ring of adèles [1]) is a central object of class field theory, a branch of algebraic number theory. It is the restricted product of all the completions of the global field and is an example of a self-dual topological ring. An adele derives from a particular ...
The restricted holonomy group based at x is the subgroup coming from contractible loops γ. If M is connected, then the holonomy group depends on the basepoint x only up to conjugation in GL(k, R). Explicitly, if γ is a path from x to y in M, then
Let C[0,1] denote the Banach space of continuous real-valued functions on the interval [0,1] with the sup norm. Denote the subspace of all functions f ∈ C[0,1] with f(0) = 0 by M. Then the equivalence class of some function g is determined by its value at 0, and the quotient space C[0,1]/M is isomorphic to R.
The concept of projection in mathematics is a very old one, and most likely has its roots in the phenomenon of the shadows cast by real-world objects on the ground. This rudimentary idea was refined and abstracted, first in a geometric context and later in other branches of mathematics. Over time different versions of the concept developed, but ...
For an associative algebra A over a field k of characteristic p>0, the commutator [,]:= and the p-mapping []:= make A into a restricted Lie algebra. [1] In particular, taking A to be the ring of n x n matrices shows that the Lie algebra () of n x n matrices over k is a restricted Lie algebra, with the p-mapping being the pth power of a matrix.