2.1.5 Tensor Abstraction Vector Space Dependence

Tensor Abstraction Vector Space Dependence is the relationship between the basis-free, abstract formulation of a tensor as a multilinear map on V and V*, and the specific underlying space V that formulation still presupposes, describing how abstraction removes dependence on any particular basis or coordinate system while leaving the dependence on V itself, its dimension, its field, fully intact, and explaining why "basis-independent" is not the same as "vector-space-independent."

What Abstraction Removes and What It Retains

Abstraction Removes Dependence on a Chosen Basis

The abstract definition of a type (p, q) tensor as a multilinear map from p copies of V* and q copies of V into the field F makes no reference to any particular basis; the map is defined by its action on arbitrary vector and covector arguments, valid regardless of which basis is later used to compute its components.

Abstraction Does Not Remove Dependence on V Itself

This basis-free definition still explicitly names V and F as its domain and codomain; changing the underlying vector space, replacing V with a different space W of a different dimension or over a different field, produces a genuinely different tensor space, T^p_q(W), unrelated to T^p_q(V) without an additional identification between V and W being supplied.

Two Distinct Kinds of Independence

Basis Independence

Basis independence means a statement or definition holds no matter which basis of the same fixed V is used; the abstract definition of a tensor, and any invariant scalar formed by fully contracting tensors, are basis independent in exactly this sense.

Vector Space Independence

Vector space independence, by contrast, would mean a statement holds regardless of which vector space is used as V altogether, a far stronger and generally false claim for concrete tensors, since a tensor's type, dimension-dependent component count, and interpretation are all tied to the specific V it was built from.

Why This Distinction Matters When Reading Abstract Notation

Abstract Index-Free Notation Can Obscure the Underlying V

Writing a tensor equation in purely abstract, index-free notation, without ever naming V explicitly, can create the impression that the statement holds for any vector space whatsoever, when in fact the statement is only claimed, and only proven, for whatever specific V was fixed at the point the tensors involved were originally defined.

Two Abstractly Identical-Looking Statements Can Concern Different Spaces

Two equations that look identical in abstract notation, T(u, v) = T(v, u) for instance, are only comparable, or combinable, if both refer to tensors built from the same underlying V; superficial notational similarity does not imply the underlying spaces are the same or even compatible.

Generalization Across Different Vector Spaces

Statements That Do Generalize

Certain structural facts genuinely do hold for any finite-dimensional V over any field, that a type (p, q) tensor has n^{p+q} components in dimension n, that a full contraction yields a type (0, 0) scalar, these are theorems about the tensor construction as a general procedure, provable once for arbitrary V and then applicable to any specific instance.

Statements That Do Not Generalize

Other facts are true only for a specific V, the numerical value of a particular tensor's components, the interpretation of a tensor as representing a specific physical quantity, or the existence of a particular metric with specific properties, none of which follow merely from the general theory and instead depend on which V and which additional structure was actually chosen.

Relationship to the Broader Vector Space Foundations

Abstraction as a Tool, Not an Escape From V

Abstract, basis-free formulations of tensor algebra are valuable precisely because they isolate which facts depend only on the general construction and which depend on a specific V, but this value is realized only by keeping clear, at every step, exactly which of the two kinds of independence, basis or vector-space, a given abstract statement is actually claiming.

Anchoring Back to the Vector Space Role

Because every tensor construction is ultimately anchored in one fixed V, as described in the vector space's broader role within tensor algebra, abstraction never removes the need to know, for any specific claim, which V is being assumed; it only removes the need to fix a specific basis of that V before the claim can be stated.