Multilinear Maps
Linear maps take a single vector argument and depend linearly on it. The constructions to come
require functions of several vector arguments, depending linearly on each one separately. A covector
is the one-argument case; the multi-argument generalization is the language of tensors, which
organizes inner products, determinants, and brackets, and supplies the algebraic substrate for
metrics and differential forms on manifolds.
Definition: Multilinear Map
Let \(V_1, \ldots, V_k\) and \(W\) be real vector spaces. A map
\(F : V_1 \times \cdots \times V_k \to W\) is multilinear if it is linear in
each argument separately: for each index \(i\), with the other arguments held fixed,
\[
F(v_1, \ldots, a v_i + a' v_i', \ldots, v_k)
= a\, F(v_1, \ldots, v_i, \ldots, v_k)
+ a'\, F(v_1, \ldots, v_i', \ldots, v_k) .
\]
A multilinear map of one argument is a linear map; a multilinear map of two arguments is called
bilinear. The set of all multilinear maps
\(V_1 \times \cdots \times V_k \to W\), with pointwise addition and scalar multiplication, is a
real vector space, denoted \(L(V_1, \ldots, V_k; W)\).
Several constructions already in use are multilinear, a fact that the present language unifies. An
inner product
on a real vector space is a bilinear function of two vectors, used to measure lengths and angles. The
determinant,
viewed as a function of the \(n\) columns of a matrix, is a real-valued multilinear function of \(n\)
vectors, vanishing exactly when they are linearly dependent. The
Lie bracket
of a Lie algebra is a bracket-valued bilinear function of two elements. These share no obvious
feature beyond multilinearity; isolating that property is what makes a single theory possible.
The case that will matter most is \(W = \mathbb{R}\): real-valued multilinear functions. The simplest
such building block is the product of two covectors, which already exhibits the pattern that the
general tensor product formalizes.
Definition: Tensor Product of Covectors
Let \(V\) be a real vector space and let \(\omega, \eta \in V^*\) be covectors. Their
tensor product is the bilinear function \(\omega \otimes \eta : V \times V \to \mathbb{R}\)
defined by
\[
\omega \otimes \eta\,(v_1, v_2) = \omega(v_1)\, \eta(v_2) ,
\]
the product on the right being ordinary multiplication of real numbers. More generally, for
covectors \(\omega^1, \ldots, \omega^k \in V^*\), the function
\(\omega^1 \otimes \cdots \otimes \omega^k : V \times \cdots \times V \to \mathbb{R}\) is
\[
\omega^1 \otimes \cdots \otimes \omega^k\,(v_1, \ldots, v_k)
= \omega^1(v_1)\, \cdots\, \omega^k(v_k) .
\]
Linearity of each covector in its single argument makes \(\omega^1 \otimes \cdots \otimes \omega^k\)
linear in each \(v_i\), so it is an element of \(L(V, \ldots, V; \mathbb{R})\). The construction
extends to multilinear functions on different spaces: given
\(F \in L(V_1, \ldots, V_k; \mathbb{R})\) and \(G \in L(W_1, \ldots, W_l; \mathbb{R})\), their
tensor product \(F \otimes G\) is the multilinear function of \(k + l\) arguments
\[
F \otimes G\,(v_1, \ldots, v_k, w_1, \ldots, w_l)
= F(v_1, \ldots, v_k)\, G(w_1, \ldots, w_l) .
\]
This operation is bilinear in \(F\) and \(G\); it is also associative, since both
\((F \otimes G) \otimes H\) and \(F \otimes (G \otimes H)\) send their arguments to the same product
\(F(\cdots)\, G(\cdots)\, H(\cdots)\) of real numbers, the associativity of which is inherited from
multiplication in \(\mathbb{R}\). Tensor products of three or more factors may therefore be written
without parentheses.
The Tensor Product and Its Universal Property
The tensor product of vector spaces has already appeared on this site, constructed
through a choice of bases:
given bases of \(V\) and \(W\), the space \(V \otimes W\) is the vector space whose basis is the set
of formal symbols \(\mathbf{e}_i \otimes \mathbf{f}_j\), and elementary tensors are defined by
bilinear expansion of their components. That construction is concrete and computational; it is the
viewpoint under which a tensor is a multidimensional array and the
Kronecker product
is the coordinate shadow of \(\otimes\) acting on linear maps. It was deferred there that a basis-free
characterization exists. This section supplies that characterization and connects it to the
multilinear-function viewpoint of the previous section.
The defining feature of the tensor product is not its basis but the way bilinear (more generally,
multilinear) maps factor through it. Every multilinear map out of a product
\(V_1 \times \cdots \times V_k\) is the same data as a single linear map out of
\(V_1 \otimes \cdots \otimes V_k\). This is the universal property, and it determines the tensor
product up to a unique isomorphism, with no basis chosen.
Theorem: Characteristic Property of the Tensor Product
Let \(V_1, \ldots, V_k\) be finite-dimensional real vector spaces. There is a vector space
\(V_1 \otimes \cdots \otimes V_k\) together with a multilinear map
\[
\pi : V_1 \times \cdots \times V_k \to V_1 \otimes \cdots \otimes V_k ,
\qquad \pi(v_1, \ldots, v_k) = v_1 \otimes \cdots \otimes v_k ,
\]
with the following property: for every real vector space \(X\) and every multilinear map
\(A : V_1 \times \cdots \times V_k \to X\), there is a unique linear map
\(\widetilde{A} : V_1 \otimes \cdots \otimes V_k \to X\) such that
\(\widetilde{A} \circ \pi = A\). The space \(V_1 \otimes \cdots \otimes V_k\) is determined by
this property up to a unique isomorphism, independently of any choice of basis.
Proof:
One constructs \(V_1 \otimes \cdots \otimes V_k\) as a quotient of the free vector space on the
set \(V_1 \times \cdots \times V_k\) by the subspace generated by all the relations that force
multilinearity of \(\pi\) — the differences
\((v_1, \ldots, a v_i, \ldots, v_k) - a\,(v_1, \ldots, v_i, \ldots, v_k)\) and
\((v_1, \ldots, v_i + v_i', \ldots, v_k) - (v_1, \ldots, v_i, \ldots, v_k) - (v_1, \ldots, v_i', \ldots, v_k)\).
Writing \(v_1 \otimes \cdots \otimes v_k\) for the class of \((v_1, \ldots, v_k)\), the map
\(\pi\) is multilinear by construction.
Given a multilinear \(A : V_1 \times \cdots \times V_k \to X\), the characteristic property of
the free vector space extends \(A\) to a unique linear map on the free space; multilinearity of
\(A\) means precisely that this linear map kills the relation subspace, so it descends to a
unique linear map \(\widetilde{A}\) on the quotient with \(\widetilde{A} \circ \pi = A\).
Uniqueness of \(\widetilde{A}\) holds because the elements \(v_1 \otimes \cdots \otimes v_k\) span
the quotient and \(\widetilde{A}\) is determined on them by
\(\widetilde{A}(v_1 \otimes \cdots \otimes v_k) = A(v_1, \ldots, v_k)\). Any two spaces with the
stated property receive, from each other's universal maps, mutually inverse linear maps, so the
tensor product is unique up to a unique isomorphism.
The same quotient construction, applied with a chosen basis, recovers exactly the basis description
used earlier: the classes \(\mathbf{e}_i \otimes \mathbf{f}_j\) form a basis of the quotient, and the
dimension is the product of the dimensions. The basis construction and the universal property
therefore describe one and the same space, the former convenient for computation, the latter for
coordinate-free reasoning.
Two viewpoints on one space
The multilinear-function viewpoint of the previous section and the abstract tensor product are linked
by a canonical isomorphism, which is what licenses treating tensors interchangeably as abstract
products of covectors or as multilinear functions.
Proposition: Abstract Versus Concrete Tensor Products
For finite-dimensional real vector spaces \(V_1, \ldots, V_k\), there is a canonical isomorphism
\[
V_1^* \otimes \cdots \otimes V_k^* \;\cong\; L(V_1, \ldots, V_k; \mathbb{R}) ,
\]
under which the abstract tensor product \(\omega^1 \otimes \cdots \otimes \omega^k\) of covectors
corresponds to the multilinear function
\((v_1, \ldots, v_k) \mapsto \omega^1(v_1) \cdots \omega^k(v_k)\).
Proof:
The assignment \((\omega^1, \ldots, \omega^k) \mapsto \big[(v_1, \ldots, v_k) \mapsto \omega^1(v_1) \cdots \omega^k(v_k)\big]\)
is multilinear in the covectors \(\omega^1, \ldots, \omega^k\), so by the universal property it
descends to a unique linear map
\(\Phi : V_1^* \otimes \cdots \otimes V_k^* \to L(V_1, \ldots, V_k; \mathbb{R})\) sending
\(\omega^1 \otimes \cdots \otimes \omega^k\) to that function. To see that \(\Phi\) is an
isomorphism it suffices to check that it is surjective, since the two spaces have equal dimension
\(\dim V_1 \cdots \dim V_k\): the domain by the dimension of a tensor product, the codomain by the
count of multilinear functions, both recorded with the coordinate basis established below.
Surjectivity is immediate, because every multilinear function is determined by its values on
tuples of basis vectors, and those values are matched by the corresponding linear combination of
products of dual basis covectors, which lies in the image of \(\Phi\). The construction of
\(\Phi\) refers only to evaluation, with no basis chosen, so the isomorphism is canonical, even
though dimension and surjectivity were verified with the aid of a basis.
Henceforth the notation \(V_1^* \otimes \cdots \otimes V_k^*\) denotes either the abstract tensor
product or the space of multilinear functions \(L(V_1, \ldots, V_k; \mathbb{R})\), whichever reading
is convenient.
Two pages, two viewpoints, one object
The tensor product is the same vector space here as in the earlier treatment, but
approached from a different side, and the two should not be conflated even as they are reconciled.
The earlier page works with a fixed basis and reads a tensor as a multidimensional array of
components; the Kronecker product and vectorization are the coordinate-level operations that
result, and the perspective is the one that matches the data structures of computational
practice. The present page characterizes the tensor product without any basis, by how multilinear
maps factor through it, and identifies the tensor product of dual spaces with the space of
multilinear functions. Neither viewpoint is more correct; they are dual descriptions. The
canonical isomorphism above makes the link precise: once a basis is fixed, the array of an
element of \(V_1^* \otimes \cdots \otimes V_k^*\) in the dual basis coincides entry for entry
with the table of values that the associated multilinear function takes on tuples of basis
vectors. On manifolds the basis-free description is the
one that survives a change of coordinates, which is why the development that follows takes the
multilinear-function viewpoint as primary.
Covariant and Contravariant Tensors
The tensors that matter most for the geometry of manifolds are those built from the dual space:
multilinear functions of several vectors. A covector is the case of one argument, and the general
case is its multilinear generalization.
Definition: Covariant Tensor
Let \(V\) be a finite-dimensional real vector space and \(k\) a positive integer. A
covariant \(k\)-tensor on \(V\) is an element of the \(k\)-fold tensor product of
the dual space,
\[
T^k(V^*) = \underbrace{V^* \otimes \cdots \otimes V^*}_{k \text{ copies}} ,
\]
equivalently a real-valued multilinear function of \(k\) vectors,
\(\alpha : V \times \cdots \times V \to \mathbb{R}\). The integer \(k\) is the rank
of \(\alpha\). By convention a \(0\)-tensor is a real number, and \(T^0(V^*) = \mathbb{R}\).
The two descriptions agree by the canonical identification of the abstract tensor product with the
space of multilinear functions established above. The rank-one case is already familiar: a covariant
\(1\)-tensor is exactly a
covector,
so \(T^1(V^*) = V^*\), and the development of covectors is the rank-one instance of the present one.
Several rank-two and higher tensors are already in use: a bilinear form is a covariant \(2\)-tensor,
an
inner product
is a covariant \(2\)-tensor that is in addition symmetric and positive definite, and the
determinant,
as a function of the \(n\) columns, is a covariant \(n\)-tensor on \(\mathbb{R}^n\).
Reversing the roles of \(V\) and \(V^*\) gives the dual notion, multilinear functions of covectors
rather than vectors.
Definition: Contravariant Tensor
Let \(V\) be a finite-dimensional real vector space and \(k\) a positive integer. A
contravariant \(k\)-tensor on \(V\) is an element of the \(k\)-fold tensor product
of \(V\) with itself,
\[
T^k(V) = \underbrace{V \otimes \cdots \otimes V}_{k \text{ copies}} .
\]
Since \(V\) is finite-dimensional, \(T^k(V)\) may equivalently be regarded as the space of
real-valued multilinear functions of \(k\) covectors. In particular \(T^1(V) = V\), and by
convention \(T^0(V) = \mathbb{R}\).
The identification of \(T^k(V)\) with multilinear functions of covectors uses the canonical
isomorphism \(V \cong V^{**}\) available in finite dimensions: a vector, regarded as an element of the
double dual, is a linear function of covectors, and tensor products of vectors are multilinear
functions of covectors. Combining the two variances yields the general mixed tensor.
Definition: Mixed Tensor
Let \(V\) be a finite-dimensional real vector space and \(k, l\) nonnegative integers. A
mixed tensor of type \((k, l)\) on \(V\) is an element of
\[
T^{(k,l)}(V) = \underbrace{V \otimes \cdots \otimes V}_{k \text{ copies}}
\otimes \underbrace{V^* \otimes \cdots \otimes V^*}_{l \text{ copies}} ,
\]
with \(k\) contravariant and \(l\) covariant slots. The covariant and contravariant tensors are
the special cases
\[
T^{(0,l)}(V) = T^l(V^*), \qquad T^{(k,0)}(V) = T^k(V),
\]
and \(T^{(0,0)}(V) = \mathbb{R}\), \(T^{(0,1)}(V) = V^*\), \(T^{(1,0)}(V) = V\).
A note on conflicting notation
The placement of the indices \(k\) and \(l\) in the symbol \(T^{(k,l)}(V)\), and the use of
superscripts versus subscripts for the rank, are not universal across the literature. Some
sources reverse the roles of the two integers, and some write a single superscript or subscript
for the covariant rank. The convention adopted here puts the contravariant rank first and uses
\(T^k(V^*)\) for purely covariant tensors. The development that follows is concerned almost
entirely with covariant tensors, since a metric, a differential form, and the other structures to
come are all covariant; contravariant and mixed tensors are recorded here for completeness and
will reappear when raising and lowering indices becomes available.
Coordinate Representation
A choice of basis for \(V\) produces a basis for every tensor space over \(V\), built from tensor
products of the
dual basis
covectors. The underlying principle is that a multilinear function is determined by its values on all
tuples of basis vectors, and those values may be prescribed freely.
Theorem: Basis for the Space of Covariant Tensors
Let \(V\) be a real vector space of dimension \(n\) with basis \((E_1, \ldots, E_n)\) and dual
basis \((\varepsilon^1, \ldots, \varepsilon^n)\). For each positive integer \(k\), the set
\[
\big\{ \varepsilon^{i_1} \otimes \cdots \otimes \varepsilon^{i_k}
: 1 \leq i_1, \ldots, i_k \leq n \big\}
\]
is a basis for \(T^k(V^*)\), which therefore has dimension \(n^k\).
Proof:
Evaluating a tensor product of dual basis covectors on a tuple of basis vectors gives, by the
definition of the tensor product of covectors,
\[
\varepsilon^{i_1} \otimes \cdots \otimes \varepsilon^{i_k}\,(E_{j_1}, \ldots, E_{j_k})
= \varepsilon^{i_1}(E_{j_1}) \cdots \varepsilon^{i_k}(E_{j_k})
= \delta^{i_1}_{j_1} \cdots \delta^{i_k}_{j_k} ,
\]
which is \(1\) when the multi-index \((i_1, \ldots, i_k)\) equals \((j_1, \ldots, j_k)\) and
\(0\) otherwise. Given any covariant \(k\)-tensor \(\alpha\), set
\(\alpha_{i_1 \cdots i_k} = \alpha(E_{i_1}, \ldots, E_{i_k})\). The tensor
\(\alpha_{i_1 \cdots i_k}\, \varepsilon^{i_1} \otimes \cdots \otimes \varepsilon^{i_k}\) takes the
same value as \(\alpha\) on every tuple of basis vectors, by the displayed evaluation, and two
multilinear functions agreeing on all tuples of basis vectors are equal. Hence the displayed set
spans \(T^k(V^*)\). For independence, a vanishing combination
\(c_{i_1 \cdots i_k}\, \varepsilon^{i_1} \otimes \cdots \otimes \varepsilon^{i_k} = 0\), evaluated
on \((E_{j_1}, \ldots, E_{j_k})\), yields \(c_{j_1 \cdots j_k} = 0\) for every multi-index. There
are \(n^k\) such basis elements, so \(\dim T^k(V^*) = n^k\).
The computation in the proof records the coordinate representation of an arbitrary covariant tensor.
Definition: Components of a Covariant Tensor
With a basis \((E_i)\) and dual basis \((\varepsilon^i)\) fixed, every covariant \(k\)-tensor
\(\alpha \in T^k(V^*)\) has a unique expansion
\[
\alpha = \alpha_{i_1 \cdots i_k}\, \varepsilon^{i_1} \otimes \cdots \otimes \varepsilon^{i_k} ,
\qquad
\alpha_{i_1 \cdots i_k} = \alpha(E_{i_1}, \ldots, E_{i_k}) .
\]
The \(n^k\) scalars \(\alpha_{i_1 \cdots i_k}\) are the components of \(\alpha\)
with respect to the basis. All indices are subscripts, consistent with the covariant nature of
the tensor.
The rank-two case ties the abstract development back to matrices. A bilinear form
\(\beta \in T^2(V^*)\) has components \(\beta_{ij} = \beta(E_i, E_j)\), so
\(\beta = \beta_{ij}\, \varepsilon^i \otimes \varepsilon^j\), and the components form an
\(n \times n\) matrix \((\beta_{ij})\). The value of \(\beta\) on a pair of vectors is the bilinear
expression \(\beta(v, w) = \beta_{ij}\, v^i w^j\), the familiar matrix-vector pairing. A covariant
\(2\)-tensor is exactly a bilinear form, and its components are exactly its matrix; the symmetric and
positive-definite ones among them are the inner products, which will reappear as the pointwise data
of a metric.
The same construction applies verbatim to the contravariant and mixed tensors, using the basis
\((E_i)\) of \(V\) alongside the dual basis. A mixed tensor of type \((k, l)\) has components carrying
\(k\) upper and \(l\) lower indices, the upper indices contravariant and the lower covariant, with the
index convention again ensuring that every contraction pairs one upper index against one lower index.
When these constructions are carried out fiberwise over a manifold, with the dual coordinate basis
fields in place of \((\varepsilon^i)\), the result is the component representation of a tensor field,
the object that the next development builds on.