2.3.1 Tensor Vector Space Carrier Set

Tensor Vector Space Carrier Set is the bare underlying set of elements that a vector space V consists of before any of its vector space structure, addition, scalar multiplication, the zero element, is imposed upon it, distinguishing the raw collection of objects being called "vectors" from the algebraic operations and axioms that turn that collection into a vector space in the first place, and by extension into a legitimate foundation for a tensor construction.

The Carrier Set as Distinct From the Structure Placed on It

A Set Alone Is Not Yet a Vector Space

Any set of objects, ordered tuples of numbers, functions on an interval, geometric arrows anchored at a point, can serve as a carrier set, but none of them constitutes a vector space until specific operations of addition and scalar multiplication are defined on it and shown to satisfy the vector space axioms; the same underlying set can, in principle, support more than one distinct vector space structure.

Why This Distinction Matters for Tensor Algebra

Because tensor algebra is built entirely on the operations and axioms of the vector space structure, not merely on the existence of a set of objects, confirming that a candidate carrier set has actually been equipped with a valid vector space structure, rather than merely resembling one informally, is a prerequisite that precedes any tensor construction.

Examples of the Same Carrier Set With Different Structures

Ordered Tuples as a Carrier Set

The set of ordered n-tuples of real numbers can be given the familiar componentwise addition and scalar multiplication, producing the standard vector space ℝ^n, but the identical underlying set of tuples could instead be given a different, non-standard pair of operations, producing a structurally different vector space over the same carrier set.

Functions as a Carrier Set

A set of real-valued functions on a fixed interval can serve as a carrier set for a vector space using ordinary pointwise addition and scalar multiplication, and the resulting vector space, once shown to be finite-dimensional in a suitably restricted case, can equally well serve as the underlying V for a tensor construction, despite its elements looking nothing like the arrows or tuples more commonly pictured as vectors.

Verifying That a Carrier Set Supports a Valid Structure

Checking Closure Under the Proposed Operations

Before treating a candidate carrier set as a vector space, its proposed addition and scalar multiplication must be checked for closure, confirming that combining two elements of the set under these operations always produces another element still belonging to the same set, rather than escaping it.

Checking the Remaining Vector Space Axioms

Beyond closure, the proposed operations must be checked against the full list of vector space axioms, associativity, existence of a zero element, existence of additive inverses, distributivity over scalar multiplication, before the carrier set together with these operations can be accepted as a genuine vector space.

The Carrier Set's Role in Fixing Dimension and Basis

Elements of the Carrier Set Become Basis Candidates

Once a valid vector space structure has been confirmed, specific elements of the carrier set are selected to serve as a basis, and it is only relative to this selection, made from within the carrier set itself, that the dimension n and the component representation of every vector become defined.

The Carrier Set Does Not by Itself Determine a Preferred Basis

Because the carrier set is simply the raw collection of elements, it carries no preferred or distinguished basis on its own; any particular basis is an additional choice made after the vector space structure is fixed, not an intrinsic feature of the carrier set itself.

Why Distinguishing the Carrier Set From the Full Structure Matters

Preventing Confusion Between an Object and Its Structure

Explicitly separating the carrier set from the vector space structure imposed on it prevents the common confusion of treating any collection of objects that merely resembles vectors, informally addable, informally scalable, as automatically qualifying for tensor construction without the underlying operations and axioms ever having been verified.

A Starting Point Beneath the Broader Underlying Structure

Because the carrier set is the most basic layer beneath the broader underlying vector space structure, field, dimension, basis, and dual basis all built on top of it, recognizing it as a distinct, separately verifiable component is the first step in establishing that this entire broader structure is legitimately in place before any tensor is built upon it.