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The subspace V × {0} of V ⊕ W is isomorphic to V and is often identified with V; similarly for {0} × W and W. (See internal direct sum below.) With this identification, every element of V ⊕ W can be written in one and only one way as the sum of an element of V and an element of W. The dimension of V ⊕ W is equal to the sum of the ...
The direct sum is also commutative up to isomorphism, i.e. for any algebraic structures and of the same kind. The direct sum of finitely many abelian groups, vector spaces, or modules is canonically isomorphic to the corresponding direct product. This is false, however, for some algebraic objects, like nonabelian groups.
If (V,φ) and (W,ψ) are representations of (say) a group G, then the direct sum of V and W is a representation, in a canonical way, via the equation (,) = (,). The direct sum of two representations carries no more information about the group G than the two representations do individually. If a representation is the direct sum of two proper ...
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 ...
Each of the H i is included as a closed subspace in the direct sum of all of the H i. Moreover, the H i are pairwise orthogonal. Conversely, if there is a system of closed subspaces, V i, i ∈ I, in a Hilbert space H, that are pairwise orthogonal and whose union is dense in H, then H is canonically isomorphic to the direct sum of V i.
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 ...
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Let V be a representation of a group G; or more generally, let V be a vector space with a set of linear endomorphisms acting on it. In general, a vector space acted on by a set of linear endomorphisms is said to be simple (or irreducible) if the only invariant subspaces for those operators are zero and the vector space itself; a semisimple representation then is a direct sum of simple ...
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