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for v, v 1, v 2 ∈ V, w, w 1, w 2 ∈ W, and α ∈ K. The resulting vector space is called the direct sum of V and W and is usually denoted by a plus symbol inside a circle: It is customary to write the elements of an ordered sum not as ordered pairs (v, w), but as a sum v + w. The subspace V × {0} of V ⊕ W is isomorphic to V and is often ...
For an arbitrary family of groups indexed by , their direct sum [2] is the subgroup of the direct product that consists of the elements () that have finite support, where, by definition, () is said to have finite support if is the identity element of for all but finitely many . [3] The direct sum of an infinite family () of non-trivial groups ...
The group operation in the external direct sum is pointwise multiplication, as in the usual direct product. This subset does indeed form a group, and for a finite set of groups {H i} the external direct sum is equal to the direct product. If G = ΣH i, then G is isomorphic to Σ E {H i}. Thus, in a sense, the direct sum is an "internal ...
The vector space is said to be the algebraic direct sum (or direct sum in the category of vector spaces) when any of the following equivalent conditions are satisfied: The addition map S : M × N → X {\\displaystyle S:M\\times N\\to X} is a vector space isomorphism .
The vector space of complex-valued class functions of a group has a natural -invariant inner product structure, described in the article Schur orthogonality relations.Maschke's theorem was originally proved for the case of representations over by constructing as the orthogonal complement of under this inner product.
In general topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, free sum, topological sum, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology. Roughly speaking, in the ...
This applies also when E and F are linear subspaces or submodules of the vector space or module V. 2. Direct sum: if E and F are two abelian groups, vector spaces, or modules, then their direct sum, denoted is an abelian group, vector space, or module (respectively) equipped with two monomorphisms: and : such that is the internal direct sum of ...
More generally, is called the direct sum of a finite set of subgroups, …, of the map = is a topological isomorphism. If a topological group G {\displaystyle G} is the topological direct sum of the family of subgroups H 1 , … , H n {\displaystyle H_{1},\ldots ,H_{n}} then in particular, as an abstract group (without topology) it is also the ...