8.3.2 Tensor Indexed Symbol Index Set

A tensor indexed symbol index set labels and organizes indices in tensor notation, providing structure for multi-dimensional mathematical expressions.

Tensor Indexed Symbol Index Set is the underlying set of values — commonly {1, ..., n} for a finite-dimensional space, or an arbitrary index set I in more general settings — over which an index letter is understood to range, as distinct from the particular letter chosen to denote a generic element of that set. It is the formal object being referred to whenever an index is said to "range over" or "take values in" some collection, and it exists independently of whether that collection is written out as 1, ..., n or referred to abstractly by a set symbol such as I.

Distinguishing the Index Set from the Index Symbol

The Set Versus the Variable Ranging Over It

An index letter such as i is a variable name; the index set is the actual collection of values i is permitted to take. Writing i ∈ {1, ..., n} or i ∈ I makes this distinction explicit: the symbol i could be renamed to any other unused letter without changing anything about the set itself, since the set is a fixed piece of data, while the letter is merely a label used to refer to a generic member of it.

i \in {1, ..., n}

Finite Index Sets

For a tensor defined on a finite-dimensional vector space of dimension n, each index's set is the finite set of the first n positive integers (or, in some conventions, integers from 0 to n − 1); every index attached to a tensor on that space shares this same finite index set, unless a different dimension applies to a different factor in a tensor product of several distinct spaces.

Arbitrary and Infinite Index Sets

Beyond the Finite Case

In more general settings, such as when working with a vector space that is infinite-dimensional, an index's set is denoted abstractly by a set symbol such as I or J, without committing to any particular finite range; a basis {eᵢ}_{i∈I} for such a space is indexed by an arbitrary set I, which may be countably or uncountably infinite, and the tensor-algebra constructions built from such bases — spanning, linear independence — are stated in terms that make sense for any such index set.

{e_{i} \otimes f_{j} : i \in I, j \in J}

Products of Index Sets

When a tensor's several indices range over different underlying spaces — as in a tensor product of two distinct vector spaces with bases indexed by separate sets I and J — the combined family of basis elements is naturally indexed by the Cartesian product I × J, an index set built directly from the two original ones, illustrating how index sets themselves combine according to the algebraic structure of the tensor construction.

Diagram of Index Sets

Why the Index Set Must Be Specified

Component Count Depends on It

The number of components a tensor has, and the range of every sum performed under the summation convention, is entirely determined by the sizes of the index sets its indices range over; a tensor's type (p, q) together with the dimension of the relevant index sets fixes exactly how many numbers are needed to record all of its components.

Ambiguity Without a Stated Index Set

An index letter written without any indication of its index set — no stated dimension, no named abstract set — leaves an expression under-specified, since neither the range of summation for a dummy index nor the number of components labeled by a free index can be determined without knowing what set that index ranges over.

Relationship to Symbol Choice and Scope

Independent of Which Letter Is Used

The index set attached to a given role — say, the set of spatial directions in a physical problem — remains the same regardless of whether the letter used to range over it is i, k, or any other unused symbol; renaming an index, whether free or dummy, never changes the index set involved, only the label used to refer to a generic element of it.

Consistency With Scope Requirements

Within the scope of a single term or equation, every occurrence of a given index letter must refer to the same index set; using one letter to range over two different sets within a single scope — for instance, the same letter standing for a spatial index in one part of an expression and a wholly different index set elsewhere in that same expression — produces an inconsistency that undermines the validity of the expression as a whole.