4.23.1 Tensor Multilinear Function Notation

Tensor Multilinear Function Notation is the specific set of symbolic conventions for writing a multilinear map as a function of several variables, covering how its domain and codomain are declared, how its several arguments are separated and grouped, how partial application is denoted, and how the space of all such functions is itself named. This notation is the starting point from which index notation and abstract tensor notation are later introduced as alternatives suited to different purposes.

Declaring Domain and Codomain

The Arrow Notation

A multilinear map is declared using the same arrow notation as an ordinary function, f: V₁ × ... × Vₙ → W, with the domain written as a Cartesian product of the argument spaces and the codomain written as the single target space. The Cartesian product in the domain, rather than a direct sum or a single combined space, signals that f accepts n separate vectors, one from each factor, rather than a single vector from some combined space.

The Semicolon Convention for the Space of Such Maps

The vector space consisting of all multilinear maps from V₁ × ... × Vₙ to W is commonly written Multilinear(V₁, ..., Vₙ; W) or L(V₁, ..., Vₙ; W), with a semicolon separating the list of domain factors from the codomain; this distinguishes the notation from Hom(V, W), the space of linear maps between two single spaces, and from the Cartesian product notation V₁ × ... × Vₙ used for the domain of an individual map f itself.

Writing the Arguments

Comma-Separated Argument Lists

Individual arguments of a multilinear map are listed, separated by commas, inside a single set of parentheses following the function symbol, f(v₁, v₂, ..., vₙ), matching the convention used for ordinary functions of several variables; this is distinct from writing several separate function applications, f(v₁)(v₂)⋯(vₙ), which instead denotes the curried form of the map.

Distinguishing Multilinear Arguments From a Single Tuple Argument

Some notational systems instead present the arguments bundled as a single tuple, f((v₁,...,vₙ)), emphasizing that f is, formally, an ordinary function whose single input happens to be an element of the Cartesian product V₁ × ... × Vₙ; multilinearity is then the added requirement that this function be linear in each coordinate of the tuple separately, a requirement not implied by treating the tuple as a single undifferentiated input.

Placeholder Notation for Partial Application

The Dash or Dot Convention

Fixing all but one argument and leaving the remaining slot open is denoted by a placeholder symbol, most commonly a dash or a centered dot, in the position of the free argument: f(v, -) or f(v, ·) denotes the linear map on the second factor obtained by fixing the first argument to v. This notation makes explicit which slot remains variable without introducing a new bound variable name.

Multiple Placeholders

When more than one slot is left open, repeated dashes or dots are used, f(v, -, -) denoting the bilinear map obtained by fixing only the first of three arguments; care is taken in such notation to ensure the reader understands that the two placeholders denote two independent, not necessarily equal, remaining arguments.

Composition and Combination Notation

Composing With a Linear Map

Post-composing a multilinear map f: V₁ × ... × Vₙ → W with a linear map h: W → U is written h ∘ f, understood to mean the multilinear map (v₁,...,vₙ) ↦ h(f(v₁,...,vₙ)); this notation directly parallels ordinary function composition and is used when discussing how properties of f are transported through h.

Precomposing With Linear Maps on Each Slot

Given linear maps φᵢ: Uᵢ → Vᵢ, the multilinear map obtained by precomposing each slot, (u₁,...,uₙ) ↦ f(φ₁(u₁),...,φₙ(uₙ)), is sometimes denoted f ∘ (φ₁ × ... × φₙ), using the Cartesian product of the individual linear maps to indicate that each slot is transformed independently before f is applied.

Notation for the Multilinearity Condition Itself

Displaying One Slot at a Time

The defining condition of multilinearity is written by displaying the varying slot explicitly and abbreviating the fixed slots with an ellipsis, f(..., αv + βv', ...) = αf(..., v, ...) + βf(..., v', ...), a notation that keeps attention on the single slot under discussion while signaling that the remaining arguments, whatever they may be, play no active role in the equation.

Explicit Slot Indexing

When precision about which slot is varying matters, the slot index is written explicitly rather than relying on an ellipsis, f(v₁,...,αvᵢ+βvᵢ',...,vₙ) = αf(v₁,...,vᵢ,...,vₙ) + βf(v₁,...,vᵢ',...,vₙ), a more cumbersome but fully unambiguous notation used in formal proofs where the position of the varying slot must be tracked exactly.

Notational Variants Across Sources

Parentheses Versus Angle Brackets

Some sources, particularly when a multilinear map is a pairing rather than a general map, use angle brackets instead of parentheses, ⟨v, w⟩ for a bilinear pairing, especially when the pairing has an inner-product-like character; this notation signals the pairing role of the map even though it is formally interchangeable with the standard parenthetical functional notation f(v,w).

Superscript Versus Subscript Argument Lists

In some treatments of tensor algebra, arguments are distinguished by whether they are vectors or covectors using superscripts and subscripts on the function symbol itself rather than purely through parenthetical argument lists, prefiguring the index notation used for component arrays, though the underlying functional notation f(v₁,...,vₙ) remains available as the coordinate-free description beneath any such stylistic variant.

Why Multiple Notational Conventions Coexist

Matching Notation to Context

Functional notation with commas is preferred when emphasizing the multilinear map as an operation to be evaluated; placeholder notation is preferred when discussing partial application or induced linear maps; composition notation is preferred when relating a multilinear map to others built from it. No single notational convention serves every purpose equally well, which is why fluency across these variants, rather than commitment to just one, is expected when reading across different treatments of multilinear maps.