Conceptual Tensor: A Coordinate-Free Approach

• Conceptual Tensor is an invariant, basis-independent multilinear object that unifies array, tensor space, and map representations.
• It links abstract definitions with practical computations via component arrays, ensuring transformation laws under basis changes.
• The concept underscores the equivalence of tensor interpretations, enhancing clarity in both theoretical analyses and applied operations.

A conceptual tensor is a mathematical object capturing multiway linear or multilinear structure, with the central unifying principle that a tensor’s meaning is best captured by its invariant, coordinate-free structure—though practical manipulation often proceeds via concrete representations such as arrays or component-wise formulas. The term “conceptual tensor” is used to contrast the abstract, basis-independent mathematical object with its coordinate (array-based) representation, and to illuminate the different interpretations and algebraic structures underpinning classical and modern tensor analysis.

1. Interpretations of the Conceptual Tensor

There are three principal interpretations of classical tensors, each offering a different conceptual viewpoint:

1. Coordinatizable Objects: Tensors as collections of components (arrays) whose transformation laws under change of basis define their tensorial nature. Here the object is recognized as a tensor if its arrays transform in a specified multilinear way. For example, for components $t^{ij}_k$, the transformation under a change of basis is determined by the associated linear maps and the variance of the indices.
2. Elements of Tensor Spaces: Tensors as intrinsic elements of spaces of the form $V^{\otimes n} \otimes (V^*)^{\otimes m}$ (or $(V^*)^{\otimes m} \otimes V^{\otimes n}$), where $V$ is a finite-dimensional vector space. An element $t$ of $(V^*)^{\otimes m} \otimes V^{\otimes n}$ is a tensor of type $(m,n)$, with $m$ covariant and $n$ contravariant indices, reflecting the number of slots accepting vectors and dual vectors, respectively. In index notation,

$$t^{b_1\ldots b_n}_{a_1 \ldots a_m} = (f_{a_1} \otimes \cdots \otimes f_{a_m}) \otimes (v_{b_1} \otimes \cdots \otimes v_{b_n})$$

where $f_{a_i} \in V^*$ and $v_{b_j} \in V$.

1. Linear Tensor Maps: Tensors as linear maps between tensor powers of vector spaces,

$T: V^{\otimes m} \rightarrow V^{\otimes n}$

identifying a tensor as a multilinear function accepting $m$ arguments from $V$ and producing an element of $V^{\otimes n}$. This interpretation is more operational and lends itself to a function-theoretic perspective: subscripts in indices correspond to argument slots, superscripts to value slots.

All three are canonically isomorphic for finite-dimensional vector spaces,

$$(V^*)^{\otimes m} \otimes V^{\otimes n} \cong L(V^{\otimes m}, V^{\otimes n})$$

where $L(V^{\otimes m}, V^{\otimes n})$ is the space of linear maps between the corresponding tensor powers of $V$.

2. Structural Roles: Array, Tensor Space, and Map Viewpoints

The distinction between tensor as an abstract invariant and as an array of components is a foundational aspect of the conceptual tensor. The array representation depends on a basis; the conceptual tensor is the equivalence class of all possible arrays related by the proper transformation law:

• Arrays or Generalized Matrices: For a given basis, the tensor is represented as a multi-index array; index positions encode the variance (covariant/contravariant character) and thus the transformation properties.
• Tensor Transformation Law: Change of basis (or coordinates) leads to component transformations that ensure the invariance of the underlying tensor; in contrast, the array itself is generally not invariant.

The conversion between the coordinate-free tensor and its array representation requires explicit knowledge of the basis and the transformation rules. Conceptually, the tensor is independent of any such representation.

3. Tensor Operations as Functionality: Generalized Composition

In the tensor-as-map viewpoint, the concept of tensor multiplication becomes function composition generalized to multilinear mappings:

• Generalized Slot Matching: Composition (or contraction) occurs where the output slots (upper indices) of one tensor coincide with the input slots (lower indices) of another, corresponding to the summation over repeated indices in Einstein notation.
• Contraction as Summation: In index notation, contractions correspond to summation over matching index positions, e.g.,

$$(S \circ T)^{d_1 \cdots d_{n+q-o}}_{a_1\cdots a_{m+p-o}}$$

where $o$ is the number of contracted indices.

• Unified View of Multiplication/Composition: Both ordinary function composition and tensor contraction are subsumed under this framework. For instance, applying a linear form (a (0,1)-tensor) to a vector (a (1,0)-tensor) yields a scalar (a (0,0)-tensor).

4. Mathematical Formalism and Equivalences

The formal equivalence of different conceptualizations is central:

• Isomorphism Between Approaches: For finite-dimensional $V$,

$$(V^*)^{\otimes m} \otimes V^{\otimes n} \cong L(V^{\otimes m}, V^{\otimes n})$$

so elements of the tensor space can always be interpreted as multilinear maps, and vice versa.

• Abstract Index Notation: Facilitates valence bookkeeping, distinguishing which slots are for arguments and which for outputs.

View of Tensors	Mathematical Model	Conceptual Paradigm
Coordinatizable objects	Arrays (multi-index matrices)	Representation (basis-dependent)
Tensor space elements ("bilateral")	$(V^*)^{\otimes m} \otimes V^{\otimes n}$	Intrinsic (coordinate-free)
Linear tensor maps	$L(V^{\otimes m}, V^{\otimes n})$	Functional/operational

5. Significance for Mathematics, Physics, and Practice

The shift from array-centric definitions to the conceptual tensor as a coordinate-free multilinear object is foundational for:

• Invariant Structures: Physical laws and geometric objects must be independent of the observer’s coordinate system; tensors, in their conceptual form, provide such invariance.
• Operational Clarity: In applications (e.g., general relativity, continuum mechanics, or data science), utilizing the tensor-as-map viewpoint clarifies when and how contraction (summation over indices) generalizes function composition.
• Computational Efficiency: The array view is unavoidably practical for calculation, but conceptual clarity derives from separating the tensor from its representation—thereby reducing mistakes and ambiguities when analyzing transformation properties.

6. Role of Arrays and Basis Dependence

Arrays are indispensable tools for computation but are not themselves the tensor:

• Component Arrays vs. Conceptual Tensor: The array $t^{b_1\ldots b_n}_{a_1 \ldots a_m}$ in a basis represents the tensor as components; this representation changes under coordinate transformations, whereas the tensor itself remains invariant.
• Change of Basis: The transformation law for arrays under changes of basis encodes the tensor's valence and, crucially, its status as a geometrically or physically meaningful quantity.

7. Summary and Comparative Table

This unified conceptual framework enables practitioners to choose the interpretation matching the context—abstract computations, coordinate representations, or functional operations—while maintaining the central invariant structure.

Interpretation	Abstract Object	Representation
Map	$T: V^{\otimes m} \to V^{\otimes n}$	Array with indexed slots
Tensor product	$(V^*)^{\otimes m} \otimes V^{\otimes n}$	Coordinate-free vector
Array	Multidimensional array in basis	Component representation

Conclusion:

The conceptual tensor, as highlighted in the cited work (Jonsson, 2014), is best regarded as an invariant, coordinate-independent multilinear map—formally, an element of $(V^*)^{\otimes m} \otimes V^{\otimes n}$ or, equivalently, a linear map $V^{\otimes m} \rightarrow V^{\otimes n}$. Its operations are not limited to naive array manipulations but instead are grounded in the generalized composition (function-theoretic contraction) intrinsic to its structure. The explicit distinction between the tensor itself and its coordinate representations is essential for rigorous manipulation, application, and interpretation in both mathematics and theoretical physics.