2.1.3 Tensor Operation Vector Space Dependence

Tensor operations in vector spaces reveal dependencies through algebraic structures, linking linear transformations and multilinear relationships.

Tensor Operation Vector Space Dependence is the specific way in which each standard tensor operation, contraction, tensor product, symmetrization, index raising and lowering, relies on particular features of the underlying vector space V, its dimension, its field, and in some cases an additional metric structure on it, so that the validity and meaning of an operation is tied directly to properties of V rather than being a universally applicable rule independent of the space it is performed in.

Tensor Product Depends Only on V and F

No Extra Structure Beyond the Basic Inputs

The tensor product of two tensors, combining a type (p, q) tensor and a type (r, s) tensor into a type (p+r, q+s) tensor, depends only on V and its field F as already fixed by the construction, requiring no additional structure such as a metric or an inner product.

T_{q}^{p} \otimes S_{s}^{r} \in T_{q + s}^{p + r} (V)

Why This Operation Is the Least Demanding

Because tensor product simply combines existing multilinear maps into a larger one without requiring vectors and covectors to be compared or interconverted, it is the operation least dependent on any special feature of V beyond its basic status as a finite-dimensional vector space over a field.

Contraction Depends on the Pairing Between V and V*

Using the Built-In Evaluation Pairing

Contraction relies specifically on the natural pairing between V and V*, a covector evaluated on a vector, which exists automatically as soon as V* is defined as the dual space, requiring no additional structure on V itself.

T^{i j} S_{j} = contraction using the natural pairing of V and V^{*}

Why Contraction Requires One Upper and One Lower Index Specifically

Because the natural pairing exists only between V and V*, contraction is defined only between an upper (covector-consuming) slot and a lower (vector-consuming) slot; there is no analogous natural pairing between two upper indices or two lower indices without additional structure, which is exactly the dependence that makes contraction of two indices of the same variance require a metric.

Index Raising and Lowering Depends on a Chosen Metric

An Operation Not Available From V Alone

Raising or lowering an index converts between a vector in V and a covector in V*, an operation that requires a specific choice of metric, a nondegenerate bilinear form on V, since V alone, without such a form, provides no canonical way to associate a particular covector to a particular vector.

v_{i} = g_{i j} v^{j}

Different Metrics on the Same V Give Different Results

Because this operation depends on the specific metric chosen, two different metrics defined on the identical space V will in general convert the same vector into two different covectors, so raising and lowering indices is not intrinsic to V alone but to V together with a specifically chosen additional structure.

Symmetrization and Antisymmetrization Depend on the Field's Characteristic

An Assumption Usually Left Implicit

Symmetrization and antisymmetrization typically involve dividing by the factorial of the number of indices being permuted, an operation that requires the field F to allow division by that integer, which fails if F has a characteristic dividing that factorial.

T_{(i j)} = \frac{1}{2} (T_{i j} + T_{j i})

Why This Dependence Is Usually Invisible

Over the real or complex numbers, the fields almost universally used in applied tensor algebra, this division is always well defined, which is why the field-dependence of symmetrization is rarely made explicit, but it becomes a genuine restriction the moment tensor algebra is developed over a field of finite characteristic.

Summary of Dependence Across Operations

Ranking Operations by How Much Extra Structure They Require

Tensor product requires only V and F; contraction requires additionally the natural V-V* pairing already built into the dual space construction; index raising and lowering requires a specifically chosen metric beyond V alone; symmetrization requires a mild condition on F's characteristic, ordering the operations from least to most dependent on structure beyond the bare vector space input.

Why Recognizing This Ordering Matters

Recognizing which operations depend only on V and F and which require an additionally chosen metric prevents a common confusion, treating index raising and lowering as though it were as intrinsic to V as contraction is, when in fact it silently presupposes an extra structure that must be specified before the operation is even well defined.