tensor product (vector spaces)

Definition. The classical conception of the tensor productoperation involved finite dimensional vector spaces $A$, $B$, say overa field $𝕂$. To describe the tensor product $A\otimes B$ onewas obliged to chose bases

$${𝐚}_i\in A,i\in I,{𝐛}_j\in B,j\in J$$

of $A$ and $B$ indexedby finite sets $I$ and $J$, respectively, and represent elements of$𝐚\in A$ and $𝐛\in B$ by their coordinates relative to these bases,i.e. as mappings $a:I\to 𝕂$ and $b:J\to 𝕂$such that

$$𝐚=\sum _{i\in I}{a}^i{𝐚}_i,𝐛=\sum _{j\in J}{b}^j{𝐛}_j.$$

One then represented $A\otimes B$ relative to this particular choiceof bases as the vector space of mappings $c:I\times J\to 𝕂$. These mappings were called “second-order contravarianttensors” and their values were customarily denoted by superscripts,a.k.a. contravariant indices:

$$c^{ij}\in 𝕂,i\in I,j\in J.$$

The canonical bilinearmultiplication (also known as outer multiplication)

$$\otimes :A\times B\to A\otimes B$$

was defined byrepresenting $𝐚\otimes 𝐛$, relative to the chosen bases, as thetensor

$$c^{ij}=a^i{b}^j,i\in I,j\in J.$$

In this system, theproducts

$${𝐚}_i\otimes {𝐛}_j,i\in I,j\in J$$

were represented by basictensors, specified in terms of the Kronecker deltas as the mappings

$$(i^{\prime },j^{\prime })\mapsto {\delta }_i^{i^{\prime }}{\delta }_j^{j^{\prime }},i^{\prime }\in I,j^{\prime }\in J.$$

These gave a basis of $A\otimes B$.

The construction is independent of the choice of bases in thefollowing sense. Let

$${𝐚}_i^{\prime }\in A,i\in I^{\prime },{𝐛}_j^{\prime }\in B,j\in J^{\prime }$$

be different bases of $A$ and $B$ with indexing sets $I^{\prime }$and $J^{\prime }$ respectively. Let

$$r:I\times I^{\prime }\to 𝕂,s:J\times J^{\prime }\to 𝕂,$$

be the corresponding change of basis matrices determined by

${𝐚}_{i^{\prime }}^{\prime }$$={\displaystyle \sum _{i\in I}}\left(r_{i^{\prime }}^i\right){𝐚}_i,i^{\prime }\in I^{\prime }$

${𝐛}_{j^{\prime }}^{\prime }$$={\displaystyle \sum _{j\in I}}\left(s_{j^{\prime }}^j\right){𝐛}_j,j^{\prime }\in J^{\prime }.$

One then stipulated that tensors $c:I\times J\to 𝕂$ and$c^{\prime }:I^{\prime }\times J^{\prime }\to 𝕂$ represent the same element of$A\otimes B$ if

$$c^{ij}=\sum _{\genfrac{}{}{0.0pt}{}{i^{\prime }\in I^{\prime }}{j^{\prime }\in J^{\prime }}}\left(r_{i^{\prime }}^i\right)\left(s_{j^{\prime }}^j\right){(c^{\prime })}^{i^{\prime }{j}^{\prime }}$$(1)

for all $i\in I,j\in J$. This relation corresponds to the fact thatthe products

$${𝐚}_i^{\prime }\otimes {𝐛}_j^{\prime },i\in I^{\prime },j\in J^{\prime }$$

constitute an alternate basis of $A\otimes B$, and that the change ofbasis relations are

$${𝐚}_{i^{\prime }}^{\prime }\otimes {𝐛}_{j^{\prime }}^{\prime }=\sum _{\genfrac{}{}{0.0pt}{}{i\in I}{j\in J}}\left(r_{i^{\prime }}^i\right)\left(s_{j^{\prime }}^j\right){𝐚}_i\otimes {𝐛}_j,i^{\prime }\in I^{\prime },j^{\prime }\in J^{\prime }.$$(2)

Notes. Historically, the tensor product was called the outer product, and has its origins in the absolute differentialcalculus (the theory of manifolds). The old-time tensor calculus isdifficult to understand because it is afflicted with a particularlylethal notation that makes coherent comprehension all but impossible.Instead of talking about an element $𝐚$ of a vector space, one wasobliged to contemplate a symbol ${𝐚}^i$, which signified a list ofreal numbers indexed by $1,2,\mathrm{\dots },n$, and which was understood torepresent $𝐚$ relative to some specified, but unnamed basis.

What makes this notation truly lethal is the fact a symbol ${𝐚}^j$ wastaken to signify an alternate list of real numbers, also indexed by$1,\mathrm{\dots },n$, and also representing $𝐚$, albeit relative to adifferent, but equally unspecified basis. Note that the choice ofdummy variables make all the difference. Any sane system ofnotation would regard the expression

$${𝐚}^i,i=1,\mathrm{\dots },n$$

as representing a list of $n$ symbols

$${𝐚}^1,{𝐚}^2,\mathrm{\dots },{𝐚}^n.$$

However, in the classical system, one was strictly forbidden fromusing

$${𝐚}^1,{𝐚}^2,\mathrm{\dots },{𝐚}^n$$

because where, after all, is the all importantdummy variable to indicate choice of basis?

Thankfully, it is possible to shed some light onto this confusion (Ihave read that this is credited to Roger Penrose) by interpreting thesymbol ${𝐚}^i$ as a mapping from some finite index set $I$ to$ℝ$, whereas ${𝐚}^j$ is interpreted as a mapping from anotherfinite index set $J$ (of equal cardinality) to $ℝ$.

My own surmise is that the source of this notational difficulty stemsfrom the reluctance of the ancients to deal with geometric objectsdirectly. The prevalent superstition of the age held that in order tohave meaning, a geometric entity had to be measured relative tosome basis. Of course, it was understood that geometrically no onebasis could be preferred to any other, and this leads directly to thedefinition of geometric entities as lists of measurements modulo theequivalence engendered by changing the basis.

It is also worth remarking on the contravariant nature of therelationship between the actual elements of $A\otimes B$ and thecorresponding representation by tensors relative to a basis — compareequations (1) and (2).This relationship is the source of theterminology “contravariant tensor” and “contravariant index”, andI surmise that itis this very medieval pit of darkness and confusion that spawned thepresent-day notion of “contravariant functor”.

References.

1. Levi-Civita, “The Absolute Differential Calculus.”