2.1.2 Tensor Construction Vector Space Input

Tensor Construction Vector Space Input is the precise data that must be supplied before the construction of a tensor space of a given type can proceed, the vector space V itself, its field of scalars F, and a target type (p, q), treated as the fixed inputs from which the construction produces a new space of multilinear maps, without yet requiring any particular basis, metric, or additional structure beyond what these inputs specify.

The Three Required Inputs

The Vector Space Itself

The first required input is the vector space V in its entirety, not merely its dimension, since the actual elements of V and the algebraic operations defined on them are what the tensor construction ultimately builds upon.

The Field of Scalars

The second required input is the field F over which V is defined, since the values a tensor's fully contracted scalar components can take, and the arithmetic available for combining them, are determined entirely by this field.

The Target Type

The third required input is the pair of non-negative integers (p, q) specifying how many copies of V* and how many copies of V the desired tensor space is to be built from, fixing in advance what kind of multilinear map the construction will produce.

What the Construction Produces From These Inputs

The Space of Type (p, q) Tensors

Given these three inputs, the construction produces a new vector space, denoted T^p_q(V), consisting of all multilinear maps taking p covector arguments and q vector arguments and returning a scalar in F.

The Output Depends Only on the Inputs Given

Two different type specifications (p, q) applied to the same V and F yield two genuinely different tensor spaces, and conversely the same type applied to two different underlying spaces V yields spaces that are not naturally identified with one another, underscoring that all three inputs jointly, not any one alone, determine the resulting tensor space.

What Is Not Required Among the Inputs

No Basis Is Required to Define the Space

The tensor space T^p_q(V) is fully and correctly defined as soon as V, F, and (p, q) are specified; a basis is needed only afterward, to represent individual tensors as component arrays, and is not itself part of the input to the construction.

No Metric Is Required for the Basic Construction

The basic construction of T^p_q(V) requires no metric or inner product on V; an additional metric input becomes necessary only for later operations, such as raising or lowering an index, that convert between V and V* beyond what the type (p, q) alone specifies.

Consequences of Varying Each Input

Varying the Type With V and F Fixed

Holding V and F fixed while varying (p, q) produces the entire family of tensor spaces built from that one underlying space, vectors, covectors, linear maps, and all higher mixed types, each a distinct member of the same family sharing the same V and F.

Varying V or F With the Type Fixed

Holding (p, q) fixed while changing V, or changing the field F from the reals to the complex numbers, for instance, produces a tensor space of the same type built on a genuinely different foundation, with no natural identification between the two resulting spaces unless one is additionally supplied.

Why Isolating the Input Explicitly Matters

Preventing Implicit or Unstated Assumptions

Stating the three required inputs explicitly guards against silently assuming a particular basis, a particular field, or a particular type when working with a tensor, each of which must instead be read off from, or explicitly declared alongside, the object under discussion.

A Starting Point for Every Subsequent Tensor Operation

Because every later operation, contraction, transformation, symmetrization, is defined relative to the specific V, F, and type from which a tensor space was built, correctly identifying these three inputs at the outset is the necessary first step before any subsequent tensor-algebraic reasoning can be applied correctly.