Direct product/Direct sum/Introduction/Section

Recall that, for a family ${}M_{i}$, ${}i\in I$, of sets ${}M_{i}$, the product set ${}\prod _{i\in I}M_{i}$ is defined. If all ${}M_{i}=V_{i}$ are ${}K$-vector spaces over a field ${}K$, then this is, using componentwise addition and scalar multiplication, again a ${}K$-vector space. This is called the direct product of vector spaces. If it is always the same space, say ${}M_{i}=V$, then we also write ${}V^{I}$. This is just the mapping space ${}\operatorname {Map} \,{\left(I,V\right)}$.

Each vector space ${}V_{j}$ is a linear subspace inside the direct product, namely as the set of all tuples

$(x_{i})_{i\in I}{\text{ with }}x_{i}=0{\text{ for all }}i\neq j.$

The set of all these tuples that are only (at most) at one place different from ${}0$ generates a linear subspace of the direct product. For ${}I$ infinite, it is not the direct product.

Definition

Let ${}I$ denote a set, and let ${}K$ denote a field. Suppose that, for every ${}i\in I$, a ${}K$-vector space ${}V_{i}$ is given. Then the set

${}\bigoplus _{i\in I}V_{i}={\left\{(v_{i})_{i\in I}\mid v_{i}\in V_{i},\,v_{i}\neq 0{\text{ for only finitely many }}i\right\}}\,$

is called the direct sum of the ${}V_{i}$.

We have the linear subspace relation

${}\bigoplus _{i\in I}V_{i}\subseteq \prod _{i\in I}V_{i}\,.$

If we always have the same vector space, then we write ${}V^{(I)}$ for this direct sum. In particular,

${}V^{(I)}\subseteq V^{I}\,$

is a linear subspace. For ${}I$ finite, there is no difference, but for an infinite index set, this inclusion is strict. For example, ${}\mathbb {R} ^{\mathbb {N} }$ is the space of all real sequences, but ${}\mathbb {R} ^{(\mathbb {N} )}$ consists only of those sequences satisfying the property that only finitely many members are different from ${}0$. The polynomial ring ${}K[X]$ is the direct sum of the vector spaces ${}KX^{n},\,n\in \mathbb {N}$. Every ${}K$-vector space with a basis ${}v_{i}$, ${}i\in I$, is "isomorphic“ to the direct sum ${}\bigoplus _{i\in I}Kv_{i}$.