This text will present a definition of tensors that will look quite different from either of both traditional definitions (one as "Einstein summation convention" by physicists, and a quite abstract definition from category theory by mathematicians) that aims to make it more intuitive and convenient to learn, for use in physics.

For this, the tensor product E⊗F between vector spaces E and F, will be defined by duality with another vector space whose definition is very close to the notion of tensor product between spaces, but only equivalent to it in the case of finite-dimensional spaces. It thus requires a different notation. As I could not find it defined by other authors (who failed to see the importance of defining it in this general way) I had to choose a new notation. I hesitated between ⊙ and ⊠ (to not mention ⊛). You can tell me your preference, of if you know an already existing convention for it.

This text, focusing on the basic definitions, may not have complete proofs yet (would need more work and algebraic preliminaries but I have so many other subjects to work on...).

One of the main other steps that would be needed after this (and before use in physics), would be the definitions and properties of symmetric and antisymmetric tensors. I wish to find a new more convenient notation for symmetric and antisymmetric tensors, before undertaking their presentation.

Taking two pairs of dual vector spaces, (E,E') and (F,F'), we
can define the space E'⊠F' of all "continuous bilinear forms" on
E×F, that is maps from E×F to ℝ whose two currified forms map E
into F' and F into E' (every element of E defines the same map
from F to ℝ as an element of F', and vice versa). This operation
between spaces can be generally defined between any two sets of
functions E'⊂ℝ^{E} and F'⊂ℝ^{F} (see the sum of functions):

E'⊠F' = {∐f |f∈F'^{E}}⋂{^{t}∐g
|g∈E'^{F}}

= {h∈ℝ^{E×F}| (Im h^{→}⊂ F'^{ })
∧ (Im h^{←}⊂ E')}

= {h∈ℝ

This simple definition implicitly contains (is equivalent to) the requirement of bilinearity, and (in infinite-dimensional cases) continuity, of each element of E'⊠F' with respect to its two variables in E and F (this information is carried by the choice of E' as a subspace of ℝ

If E is finite dimensional then the continuity condition disappears (it is always satisfied): these maps from F to E' are all linear maps from F to E' (and the same exchanging E and F). Otherwise it is anyway the set of all continuous linear maps from F to E', for the topology naturally defined by the duality.

This construction is essentially unaffected by the replacement of E by one of its generating subsets. In particular, for any basis B⊂E, the space E' of all linear forms on E is identifiable as the space ℝ

Considering the natural map from (E'⊠F' × (E×F)) to ℝ, the above procedure makes a natural map x∈E, y∈F ↦ x⊗y ∈ E⊗F where E⊗F is the dual space to E'⊠F' generated by such x⊗y.

Thus for all t∈E'⊠F', the scalar product (x⊗y).t is defined by t(x,y).

This can be generalized to the tensor product of 3 or more vector spaces : E'⊠F'⊠G' is the set of all maps from E×F×G to ℝ whose currified forms define maps from E×F to G', from E×G to F' and from F×G to E'.

Then such an object also defines a map from E to F'⊠G' (and the same in 2 other ways).

This space E'⊠F'⊠G' is identifiable to (E'⊠F')⊠G', and in duality with a space E⊗F⊗G defined the same way as above (which is identifiable to (E⊗F)⊗G).

Let us recall how ordinary algebraic expressions are formed :

Each algebraic expression, distinguishes a main symbol and a list of other well-distinguished sub-expressions (that are other expressions) entered as data to this symbol. In particular an algebraic expression is made of operation symbols.Now similarly, each tensor expression and each tensor symbol has a specific type : a set A of some finite number n of elements (the arity), and a family of vector spaces (E

The format of the list of entries to each operation symbol is determined by the type of the operation named by this symbol: the number of entries (arity), the set that each entry must belong to (domain of each argument); a set that the result belongs to.

The whole expression also has a type ( list and nature of free variables and nature of the result), the one of the operation it defines.

The list of entries need not be always labelled by the numbers 1,...,n : any given abstract finite set A of n elements can be used instead, to label by the index i ∈ A the i-th domain E_{i}of the i-th variable.

But the same tensor symbol with arity n can be interpreted as an algebraic (operation) symbol in n+1 ways: either as an n-ary operation with arguments in E'

For example a tensor symbol with arity 3, belongs to some space E⊠F⊠G. It can be either read as an operation between E',F',G' with values in ℝ, or as an operation between E' and F' with values in G, and so on for the two other arguments.

Nullary symbols (n=0) represent scalars (in ℝ)

The type of a unary symbol or expression (n=1) directly gives the vector space it belongs to when seen as a constant (which is why we choose the convention to define the type of a tensor by the duals of the domains of its arguments).

Each tensor expression of type A is linear combination of monomial tensor expressions of type A that consist in graphs structured in the following way.

The symbols in the graph occur in bulk, without any distinguished main symbol, thus without the resulting kind of order between them.

- Vertices are occurences of tensor symbols, so we have a finite
family (T
_{v}) of symbols, where v ranges over some finite set V of occurences (vertices).

- The set of ends (half-edges) is the disjoint union of the A(T
_{v}) where v ranges over V and A(T) is the type of T. In other words, an end is an oriented pair (v,c) where c∈A(T_{v}) (it is an argument of the symbol T_{v}). - Each end (v,c) is marked by the vector space associated to c.
For example if T
_{v}∈ E⊠F⊠G then its 3 ends (v,1), (v,2),(v,3) are respectively marked by the spaces E, F and G. - Some ends are grouped by pairs, forming edges (this can be expressed as an involution on this set without fixed point); each end belongs to only one edge. The spaces marking both ends of each edge, are dual to each other.
- The other ends, called the free ends (playing the role of free
variables), are put in bijection with A, preserving the labels
(spaces).

But we need to examine the problem : do tensor expressions really make sense (give a well-defined value in that intended space E

A monomial tensor expression is a tree if between any 2 vertices
there exists one and only one path that does not repeat any vertex
(so as to forget paths that can be obviously shortened by cutting
some edges). In other words it is a graph that :

- is connected : it cannot be split without cutting some edge (existence of a path between any two vertices)
- does not contain any loop (that give different paths between some vertices). In particular it does not contain any edge with its two ends at the same vertex (which is otherwise admissible in other tensor expressions).

Indeed, every vertex, or edge, or free end, can be chosen to be seen as the main symbol of the expression (if a vertex it is an operation with values in ℝ; if an edge it is a scalar product; if a free end it is an operation symbol with vectorial value); every edge is marked by the orientation of the unique path to this "main symbol", and every vertex is interpreted as the operation symbol that follows these orientations.

All these interpretations give the same result because between any two choices of main symbol there is a path, and the result is preserved at every step of this path (from each edge to each next edge), thanks to the identities between the different algebraic interpretations of each symbol (vertex).

Any graph that is not connected is divided into several separate
connected components in a unique way.

This possibility to interpret disconnected graphs can also be expressed as follows:

This rank is also equal to the dimension of the image set of each of both maps that t defines as an element of E⊠F, from E' to F and from F' to E.

Proof: The image in F of the map defined by x

If the y

Note that the proof of this equality between expressions in E⊗F is processed in the classical concept of E⊗F (universal algebraic : as quotient of the set of formal combinations of elements of E'×F' by the relations in each of E' and F') rather than as a dual to E'⊠F'. From this we can deduce that the map from the classical E⊗F to E⊠F is injective, and therefore both definitions of E⊗F coincide. We shall identify E⊗F to its image in E⊠F.

Now let us directly define the rank of an element of E⊠F as the dimension of each of its images in E and in F (and thus cannot exceed the smallest of the dimensions of E and F); it is also the dimension of the dual vector spaces it defines in the role of a scalar product between (the quotiented) E' and F'.

An element t of E⊠F belongs to E⊗F if and only if its rank is finite. In this case, its two images A in E and B in F are in duality to each other by defining for every x ∈A and y∈B, their scalar product by x.y'=y.x' (= x'ty') for any elements y'∈F' and x'∈E' such that ty'=y and x't=x.

The choice of a basis x

Thus E⊗F = E⊠F when E or F is finite dimensional, but generally not otherwise. Anyway both E⊗F and E⊠F are duals to E'⊗F' .

Now if E has finite dimension then we have a basis of n vectors e

Let us apply the trace function to this element I of E⊗E' itself. This transforms each tensor product into a scalar product, thus giving (e

- They are multilinear with respect to each symbol (distributive
over addition, and scalar factors can be put outside):

- If a tensor symbol x satisfies x=y+z then for each occurence of x in a tensor expression the result is the sum of those obtained by replacing this occurence of x with those of y and z.
- For any scalar a, replacing an occurence of x by ax
multiplies the result by a; isolated components of the graph
mean such scalar multiplication.

- Any subgraph of a graph distinguished by taking a subset of the set of vertices, can be replaced by a single symbol equal to the monomial expression defined by that subgraph. Thus when a symbol equals a linear combination of graphs, the whole equals the same linear combination where one occurence of this symbol is replaced by each graph in the combination.
- Any edge can be replaced by a "long edge" trough the identity
symbol.

But let us show that it makes sense in all other cases, that is

- Whatever tensor expression only using finite-dimensional
spaces

- More generally, expressions that may contain infinite-dimensional spaces but that do not form any loop (every loop in the graph goes through some finite-dimensional space or some tensor of finite rank).

- at a vertex representing a tensor of finite rank, or

- at an edge labelled by a finite-dimensional space, replaced by a "long edge" through the identity element that has finite rank.

This result does not depend on the choice of decomposition. Indeed if you have 2 decompositions applied to edges (or at least that does not apply 2 different decompositions to the same vertex), then let us consider making both decompositions together (if an edge is decomposed two ways, let us see it as a long edge through 2 copies of I, and apply the decomposition to a different copy of I in both cases). Then we can verify that the result of the double decomposition equals that of each of both decompositions.

Another way of seeing it, is to consider that an element of E⊠F with finite rank is identified to an element of E⊗F, which is in duality with E'⊠F'. The use of this duality gives meaning to the expression.

This formalism provides the computation : dim(E⊗F)= (dim E) (dim F)

Any family of n vectors in a space E can be formalized as an element of ℝ

In many cases, the trick to avoid any risk of mistake is to introduce vector spaces with special names to label edges in the tensor expressions, such as "ℝ

(to be continued)

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