4.23 Tensor Multilinear Map Notation

Tensor Multilinear Map Notation is the collection of symbolic conventions used to write down multilinear maps, their component arrays, and the operations performed on them, spanning functional notation borrowed from ordinary calculus, index notation favored in physics and differential geometry, and abstract tensor notation used in coordinate-free algebraic treatments. Different notations emphasize different aspects of the same underlying object, and fluency in translating between them is necessary for reading across mathematical and physical literature.

Functional Notation

Basic Form

The most direct notation writes a multilinear map exactly as a function of several arguments, f(v₁, ..., vₙ), or, when the map is bilinear, f(v, w). This notation emphasizes the map as an operation performed on vectors and is the notation used when stating the multilinearity condition itself:

Curried and Partial Application Notation

Fixing some arguments while leaving others free is written using a dash or an explicit lambda-style binding, f(v, -) or f(v, ·), denoting the linear map obtained from f by fixing the first argument to v; this notation is common when discussing the induced map to a dual space, v ↦ f(v, ·).

Index Notation

Component Arrays With Explicit Indices

Once a basis is fixed, a multilinear map's values are written using indexed components, T_{i₁...iₙ} for scalar output or T^{k}_{i₁...iₙ} when tracking an output index as well, and a general vector's coordinates are written vⁱ, with upper indices for vector coordinates and lower indices for covector coordinates, matching the classical distinction between contravariant and covariant components.

Einstein Summation Convention

Index notation is typically paired with the Einstein summation convention, in which a repeated index appearing once as an upper index and once as a lower index is automatically summed over, so that

is understood to mean a full double sum over i and j, with the explicit summation symbol omitted; this convention shortens formulas substantially in differential geometry and physics, at the cost of requiring the reader to track which indices are being summed implicitly.

Raising and Lowering Indices

When a non-degenerate bilinear form (a metric) g_{ij} is available, index notation includes the operation of raising an index, v^i = g^{ij}v_j, and lowering an index, v_i = g_{ij}v^j, using the metric and its matrix inverse g^{ij}; this notation encodes the identification between a vector space and its dual once a metric is fixed, translating freely between covariant and contravariant descriptions of the same underlying tensor.

Abstract Tensor Notation

Coordinate-Free Index Placeholders

Abstract index notation writes indices as formal labels attached to a tensor symbol without implying any specific basis or numerical values, for instance T_{ab} denoting an abstract bilinear form, distinct from T_{ij}, which denotes actual numerical components relative to a chosen basis; this notation is used to state basis-independent identities, such as symmetry T_{ab} = T_{ba}, without committing to any coordinate system.

Tensor Product Notation

The elementary tensor built from vectors v₁, ..., vₙ is written v₁ ⊗ ... ⊗ vₙ, and a general element of the tensor product is a finite sum of such terms; this notation is used whenever a multilinear map is being discussed via its factorization through the tensor product's universal property, f̃(v₁ ⊗ ... ⊗ vₙ) = f(v₁,...,vₙ).

Notation for Symmetric and Alternating Structure

Symmetrization and Antisymmetrization Brackets

Symmetrization over a set of indices is denoted by enclosing them in parentheses, T_{(ij)} = (T_{ij} + T_{ji})/2, and antisymmetrization by enclosing them in square brackets, T_{[ij]} = (T_{ij} - T_{ji})/2, with the convention extending to any number of indices by averaging over all permutations, with a sign for the antisymmetric case; this bracket notation compactly records which indices are being symmetrized or antisymmetrized without spelling out the full averaging sum each time.

Wedge and Symmetric Product Notation

The image of v₁ ⊗ ... ⊗ vₙ under alternation is written v₁ ∧ ... ∧ vₙ, the wedge product, while the image under symmetrization is written v₁⋯vₙ or v₁ ⊙ ... ⊙ vₙ, the symmetric product; these notations build the sign or invariance behavior directly into the symbol used for combination, so that v ∧ w = -w ∧ v and v ⊙ w = w ⊙ v are read directly off the notation itself.

Notational Correspondences Across Fields

Physics Versus Coordinate-Free Mathematics

Physics literature typically favors index notation with the summation convention, treating a tensor as "an object with indices that transforms in a certain way," while coordinate-free mathematical treatments favor the functional and abstract tensor notations, treating a tensor as a multilinear map or an element of a tensor product; both describe the same objects, and translating between them, replacing an index-notation formula with its corresponding basis-free statement or vice versa, is a routine but essential skill when moving between these literatures.

Currying Notation and Hom-Space Notation

The identification of an n-ary multilinear map with a linear map into a space of (n-1)-ary multilinear maps is written f ∈ Hom(V₁, Multilinear(V₂,...,Vₙ; W)), a notation emphasizing the iterated, curried structure of multilinear maps as opposed to the flat functional notation f(v₁,...,vₙ), useful when relating multilinear maps of different arities to one another.

Choosing Notation for the Task

Structural Arguments Favor Functional and Abstract Notation

Arguments establishing universal properties, canonical isomorphisms, or basis-independent facts are clearest in functional or abstract tensor notation, since these notations avoid introducing basis-dependent artifacts that could obscure which parts of an argument are genuinely intrinsic.

Explicit Computation Favors Index Notation

Direct calculation with specific numerical vectors, verification of an identity by expansion in coordinates, or translation into a form suitable for numerical software is clearest in index notation with the summation convention, since it directly mirrors the loop-based computations that would be used to evaluate the corresponding formula on a computer.