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  "body": "\n## Definition\n\nLet $F$ be a field and let $(V_i)_{i \\in I}$ be a family of [[vector spaces|000AYPA9AJ]] over $F$.\nThe *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 $i$.\nIt is a vector space over $F$ with the componentwise operations\n$$(v_i)_i + (w_i)_i = (v_i + w_i)_i, \\qquad a (v_i)_i = (a v_i)_i.$$\n\n## Remarks\n\nFor finite $I$, the direct sum equals the [[product|0EX2HPXPJ6]] $\\prod_{i \\in I} V_i$.\nFor infinite $I$, it is a proper subspace of the product when infinitely many $V_i$ are nonzero.\nEach $V_j$ embeds in the direct sum as the families that are zero outside $j$.\nThe direct sum is the [[internal direct sum|0XYYT5KNTX]] of these images.\nThe direct sum of countably many copies of $F$ is isomorphic to $F[x]$.\n\n## Explanations\n\n### L1 human\n\nThe external direct sum builds a new space out of several spaces, side by side.\nA vector of the new space chooses one vector in each space, with only finitely many nonzero choices.\nFor two spaces, this is just the space of pairs.\n",
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  "msc": [
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  "name": "External direct sum",
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