Linear and multilinear algebra; matrix theory Definition

External direct sum

Definition

Let $F$ be a field and let $(V_i)_{i \in I}$ be a family of vector spaces over $F$ . The external direct sum $\bigoplus_{i \in I} V_i$ is the set of families $(v_i)_{i \in I}$ with $v_i \in V_i$ for each $i$ and $v_i = 0$ for all but finitely many $i$ . It is a vector space over $F$ with the componentwise operations $(v_i)_i + (w_i)_i = (v_i + w_i)_i, \qquad a (v_i)_i = (a v_i)_i.$

Remarks

For finite $I$ , the direct sum equals the product $\prod_{i \in I} V_i$ . For infinite $I$ , it is a proper subspace of the product when infinitely many $V_i$ are nonzero. Each $V_j$ embeds in the direct sum as the families that are zero outside $j$ . The direct sum is the internal direct sum of these images. The direct sum of countably many copies of $F$ is isomorphic to $F[x]$ .

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Vector space definition
External direct sum definition

Tag	`02WZB422CE`
URI	`m2://02WZB422CE`
Kind	definition
Writing	W0 stub
MSC	15A03, 16D10
Defines	External direct sum
Uses	Vector space

External direct sum

Definition

Remarks

Reading path

Further reading

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