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 $F[x]$.

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.