Direct Sum vs. Direct Product vs. Tensor Product

Direct sums, direct products, and tensor products all combine mathematical structures, but they do different jobs. The key distinction is that a direct sum or product collects vectors into tuples, whereas a tensor product combines vectors from two spaces according to bilinearity.

The difference between a direct sum and a direct product matters most for an infinite family. The difference between either of those and a tensor product is already visible in finite dimensions.

Direct product: all coordinate choices are allowed

Let $(V_i)_{i\in I}$ be a family of vector spaces over the same field. Their direct product is

$$\prod_{i\in I} V_i = \{(v_i)_{i\in I} : v_i\in V_i\text{ for every }i\}.$$

There is no restriction on how many coordinates are nonzero. Addition and scalar multiplication are performed coordinate by coordinate.

For example, if every $V_i=\mathbb{R}$ and $I=\mathbb{N}$, then the sequence $(1,1/2,1/3,\ldots)$ belongs to the product. So does any other real sequence, including an unbounded one.

Direct sum: only finitely many nonzero coordinates

The (algebraic) direct sum is the subset of the product consisting of tuples with finite support:

$$\bigoplus_{i\in I} V_i =\{(v_i)\in\prod_{i\in I}V_i : v_i=0\text{ for all but finitely many }i\}.$$

“Finite support” means that only finitely many entries are nonzero. Thus $(1,1/2,1/3,\ldots)$ is in $\prod_{i\in\mathbb{N}}\mathbb{R}$ but not in $\bigoplus_{i\in\mathbb{N}}\mathbb{R}$. The tuple $(1,0,0,\ldots)$ belongs to both.

For a finite index set, every tuple automatically has finite support. Therefore the direct sum and direct product are canonically isomorphic:

$$V_1\oplus\cdots\oplus V_n \cong V_1\times\cdots\times V_n.$$

For example, $\mathbb{R}^2\oplus\mathbb{R}^3$ and $\mathbb{R}^2\times\mathbb{R}^3$ both consist of pairs $(x,y)$ with $x\in\mathbb{R}^2$ and $y\in\mathbb{R}^3$. In either case the resulting space has dimension $2+3=5$.

The direct sum also appears as an internal direct sum: a space is the direct sum of subspaces when each vector can be written uniquely as a sum of vectors from those subspaces. This describes how the subspaces fit together inside a larger space.

Tensor product: bilinear combinations

For vector spaces $V$ and $W$ over a field $\mathbb{F}$, the tensor product $V\otimes_{\mathbb{F}}W$ is a vector space built to represent bilinear operations on pairs $(v,w)$. The notation $v\otimes w$ denotes a pure tensor, and the construction enforces relations such as

$$(v_1+v_2)\otimes w=v_1\otimes w+v_2\otimes w, \qquad v\otimes(w_1+w_2)=v\otimes w_1+v\otimes w_2.$$

A tensor product is not simply the set of pairs $V\times W$. Its elements are finite sums of pure tensors, such as $v_1\otimes w_1+v_2\otimes w_2$. Different sums can represent the same tensor because of the bilinearity relations. The field (or, more generally, the base ring) matters, so it is often written as a subscript.

For finite-dimensional vector spaces, dimensions multiply:

$$\dim(V\otimes_{\mathbb{F}}W)=\dim(V)\,\dim(W).$$

So if $V=\mathbb{R}^2$ and $W=\mathbb{R}^3$, their tensor product has dimension $2\cdot3=6$, unlike their direct sum, which has dimension $2+3=5$.

Worked example: a tensor as a matrix

Let $e_1,e_2$ be the standard basis of $\mathbb{R}^2$, and let $f_1,f_2,f_3$ be the standard basis of $\mathbb{R}^3$. The six tensors $e_i\otimes f_j$ form a basis of $\mathbb{R}^2\otimes\mathbb{R}^3$.

For example, take $x=(1,2)$ and $y=(3,4,5)$. Expanding by bilinearity gives

$$\begin{aligned} x\otimes y &=(e_1+2e_2)\otimes(3f_1+4f_2+5f_3)\\ &=3e_1\otimes f_1+4e_1\otimes f_2+5e_1\otimes f_3\\ &\quad+6e_2\otimes f_1+8e_2\otimes f_2+10e_2\otimes f_3. \end{aligned}$$

Using the ordered basis $(e_1\otimes f_1,e_1\otimes f_2,e_1\otimes f_3,e_2\otimes f_1,e_2\otimes f_2,e_2\otimes f_3)$, its coordinates are $(3,4,5,6,8,10)$. Equivalently, tensors in this example can be represented by $2\times3$ matrices, with $x\otimes y$ corresponding to

$$xy^{\mathsf T}=\begin{pmatrix}3&4&5\\6&8&10\end{pmatrix}.$$

Not every tensor must be a single pure tensor. For instance, $e_1\otimes f_1+e_2\otimes f_2$ corresponds to a matrix of rank $2$, so it cannot be written as $x\otimes y$, which would give a rank-at-most-$1$ matrix.

Quick comparison

Construction	What it contains	Finite-dimensional size
$V\oplus W$	Pairs $(v,w)$	Dimensions add
$V\times W$	The same pairs when there are finitely many factors	Dimensions add
$V\otimes W$	Bilinear combinations of vectors from $V$ and $W$	Dimensions multiply

For an infinite family, the direct sum is a subspace of the direct product: every finite-support tuple is a valid product tuple, but not every product tuple has finite support. Tensor products are not another version of either tuple construction.

Common mistakes

• Treating infinite direct sums and products as interchangeable. They agree for finite index sets, but generally differ for infinite ones.
• Thinking a tensor product consists only of pairs $v\otimes w$. Pure tensors are building blocks; general tensors can be sums of them.
• Adding dimensions for a tensor product. For finite-dimensional vector spaces over a field, use the product of the dimensions. A direct sum uses their sum.
• Ignoring the setting. In functional analysis, “direct sum” may refer to a topological or Hilbert-space construction with extra conditions. The finite-support definition here is the algebraic direct sum.