---
tag: 02WZB422CE
kind: definition
name: External direct sum
msc: [15A03, 16D10]
defines: 0S7W6EG32V
uses: [05EGHEQZAW]
sources:
  - url: https://en.wikipedia.org/wiki/Direct_sum_of_modules
    locator: Construction for two vector spaces
---

## Definition

Let $F$ be a field and let $(V_i)_{i \in I}$ be a family of [[vector spaces|000AYPA9AJ]] 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|0EX2HPXPJ6]] $\prod_{i \in I} V_i$.
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|0XYYT5KNTX]] of these images.
The direct sum of countably many copies of $F$ is isomorphic to $F[x]$.

## Explanations

### L1 human

The external direct sum builds a new space out of several spaces, side by side.
A vector of the new space chooses one vector in each space, with only finitely many nonzero choices.
For two spaces, this is just the space of pairs.