5.24.3 Tensor Product Basis Notation

Tensor Product Basis Notation is the set of conventions for naming and writing the basis elements of a tensor product space, built from bases of the individual factors, ranging from the explicit doubled form eᵢ ⊗ fⱼ, through multi-index abbreviations e_I for a combined index I = (i,j), to fully suppressed juxtaposition forms such as eᵢfⱼ used once the tensor product context is established. Because a basis of V ⊗ W is always built from bases of V and W rather than chosen independently, its notation must simultaneously track two (or more) underlying indices, and different conventions balance explicitness against brevity in different ways.

The Explicit Doubled Form

Direct Notation

The most explicit basis notation writes each basis element of V ⊗ W as eᵢ ⊗ fⱼ, directly displaying both the origin of each factor and the tensor product symbol joining them, unambiguous but visually heavier than alternatives once many indices or many factors are involved.

When Explicitness Is Preferred

This form is preferred in contexts introducing the tensor product basis for the first time, or where the two factor spaces have conceptually different roles (for instance, a "position" space and a "spin" space in physics), since the explicit ⊗ and separate index letters keep each factor's origin visually distinct.

Multi-Index Abbreviation

Combining Indices into One Symbol

Once the pairing of indices (i, j) into a single joint index I is established (as in tensor product component pairing), the basis element may be abbreviated e_I, with the understanding that I ranges over the Cartesian product of the original index sets, compressing the doubled notation into a single-index form resembling the notation for a basis of any ordinary vector space.

Advantage for Dimension Counting

Writing the basis as {e_I}_{I=1}^{mn} makes the total dimension mn immediately visible as an ordinary index range, useful whenever the internal tensor product structure of the space is not the focus of the discussion and only the overall dimension matters.

Diagram of Notational Compression

Juxtaposition Notation

Suppressing the Tensor Symbol

In sufficiently established contexts, particularly in classical tensor and index notation used in physics, the basis element eᵢ ⊗ fⱼ is written simply as eᵢfⱼ, dropping the explicit ⊗; the two index letters i and j alone convey that this is a basis element of a tensor product, since ordinary (non-tensor) notation would not carry two independent free indices in this way.

Conditions Under Which Suppression Is Safe

This suppression is unambiguous only when the reader already understands that the surrounding expressions live in a tensor product space; introducing juxtaposition notation without first establishing the tensor product context risks being misread as ordinary multiplication or an unrelated product of two separate scalar quantities.

Basis Notation for Tensor Powers

Repeated Index Lists

For the n-th tensor power V^{⊗n}, basis elements are written e_{i₁} ⊗ e_{i₂} ⊗ ... ⊗ e_{iₙ}, or abbreviated e_{i₁ i₂ ... iₙ}, listing all n indices together; this generalizes the two-factor notation directly and is standard whenever tensors of order greater than two are discussed.

Symmetrized and Antisymmetrized Basis Notation

When working within symmetric or exterior powers, basis notation is further adorned to reflect the relevant symmetry: symmetrized basis elements are sometimes written with parentheses around the indices, e_{(i₁...iₙ)}, and antisymmetrized ones with square brackets, e_{[i₁...iₙ]}, a convention indicating that the corresponding sum over permutations (with or without alternating sign) has already been applied.

Dual Basis Notation for Tensor Products

Naming the Corresponding Dual Basis

The dual basis to {eᵢ ⊗ fⱼ} is written {e^i ⊗ f^j}, using upper indices matching the classical convention for dual (covariant) basis elements, and satisfying the pairing relation (e^i ⊗ f^j)(eₖ ⊗ fₗ) = δ_{ik}δ_{jl}, extending the ordinary notation for dual bases to the tensor product setting without modification beyond doubling the index.

Significance of Basis Notation

Balancing Explicitness and Brevity

The range of basis notations, from fully explicit to fully suppressed, reflects a recurring tradeoff in tensor product notation generally: more explicit forms reduce ambiguity at the cost of visual complexity, while more compressed forms improve readability once the underlying tensor product structure is firmly established in context.

Consistency Across the Levels of Tensor Notation

Because tensor product basis notation must remain compatible with component notation, index notation, and the general symbol conventions of the tensor product, the choices made here directly shape how readable and unambiguous every subsequent computation involving components, coordinates, and multi-index arrays will be.