Abstract
The combination of an additive abelian group with a ring yields a module, and a particular type of module is a linear vector space—one of the most ubiquitous mathematical structures across all branches of physics. After defining vector spaces, the first part of this chapter describes inner products, linear functionals, Gram-Schmidt orthogonalization for both coordinate and function spaces, and Hilbert spaces. This is followed by a discussion that parses the sometimes-blurry distinctions among the various types of sums and products for vector spaces. Tensors and tensor spaces are introduced, with the focus in this chapter on the metric tensor; antisymmetric tensors are discussed in Chap. 7. The chapter concludes with a discussion of cosets and quotient spaces.
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Notes
- 1.
I have assumed you are familiar with vector algebra as described in these first two paragraphs. If a short practice session is needed, see Problem 4.1.
- 2.
We often shorten this even further to just “space.” Once we get to our discussion of topology, “space” will have a different and broader meaning.
- 3.
Paul Adrien Maurice Dirac, (1902–1984), Swiss/British physicist. The origin of the terms “bra” and “ket” is a pun. Taken together in a scalar, or inner, product they form a “bracket” symbol such as \(\langle u\vert v\rangle \). See Sect. 4.3
- 4.
Note that when we multiply two real numbers in the vector space \({\mathbb {R}}^1\), or two complex numbers in the vector space \({\mathbb {C}}^1\) (see Example 4.3 (1) and (4) below), one factor is assigned to G and the other is assigned to F.
- 5.
- 6.
- 7.
- 8.
This will be important when we discuss Hilbert spaces in Sect. 4.4.4
- 9.
For notational clarity in this example, we use bold-faced letters rather than kets to represent vectors, and coordinate subscripts rather than numerical superscripts to distinguish components.
- 10.
We use the term “coordinate space” to mean that the basis vectors in the space may be written in a standard coordinate basis. We contrast this with function spaces, where the basis vectors are normalized functions.
- 11.
The concept of a dual space appears in many different guises across mathematics and physics. In the present context and because of the complex conjugation bijection, the space \(G^*\) is sometimes called a conjugate space to G. However, this term is sometimes used to denote dual spaces generally, regardless of whether complex conjugation is involved.
- 12.
In this text we use the “conjugation convention” as adopted by physicists. Mathematicians tend to reverse the conjugation between the bra and ket vectors.
- 13.
Anti-symmetric combinations of one-forms are the objects of study in exterior algebra and exterior calculus We will return to this subject in Chap. 7
- 14.
The Latin prefix sesqui- means “one-half more, half again as much.” A sesquilinear map, therefore, is “more than linear,” but not so much as to be bilinear. Other terms used to describe this map are conjugate bilinear and hermitian bilinear.
- 15.
Terminology varies slightly among authors. We follow the convention in [6], pp. 10–11.
- 16.
- 17.
- 18.
Jörgen Pedersen Gram (1850–1916), Danish number theorist and analyst; Erhard Schmidt (1876–1959), German analyst.
- 19.
The special functions arise as solutions to differential equations that are solved using integrating factors, which then give the weight functions w(x). See, for example, the treatment in [1]
- 20.
Alternative theories of integration are beyond the scope of this text, but see [20]) as a starting point if you are interested in exploring them.
- 21.
- 22.
If the inner product is strictly positive (Definition 4.6) the space is often called a pre-Hilbert space .
- 23.
- 24.
- 25.
See the comments above regarding the sum \(W = U + V\). As with all terms and symbols, it is always a good idea to double check the definitions used (and the conditions assumed) in any given text or article.
- 26.
Just think of Newton’s second law in Cartesian vs. spherical coordinates—the same physics, but two very different appearances.
- 27.
Relatedly, a particular choice of a coordinate system can cause one to conclude the presence or absence of a particular phenomenon, when in fact the “phenomenon” is nothing more than a coordinate effect. A famous example of this pertains to the event horizon of a black hole. For many years it was believed that a singularity occurred at the event horizon (within which nothing is visible to an outside observer). In fact, the event horizon “singularity” was just a coordinate effect, and the only “real” singularity is at the center of the black hole.
- 28.
Further, the tensor product \({\mathbf {u}} \otimes {\mathbf {v}}\) should not be confused with the more familiar cross-product (\(\times \)) of two vectors. For one thing, the associative property does not hold for the cross-product; this is something you may already know from your earlier study of vectors, but we will see this when we discuss algebras in Chap. 5. More fundamentally, and in terms of structure, the cross-product \({\mathbf {w}} = {\mathbf {u}} \times {\mathbf {v}}\) yields a vector, with all three vectors being in the same vector space. This is to be contrasted with the tensor product, which yields tensors defined in a different (tensor) space from the vector spaces involved in the construction.
- 29.
For ket vectors, write the basis vector index as a subscript \(({\hat{\mathbf {e}}}_i)\) and the component index as a superscript \((u^i)\). Bra vectors use the opposite positional pattern for indices. The summation convention specifies that we sum over repeated indices, but only when one index is “up” and the other is “down.” When applied “internally” to a single term \((S^{ij}_i)\) it is called a contraction of tensor indices. We have used this convention previously in the text without calling it as such.
- 30.
The origin of these terms relates to how tensor components transform under coordinate transformations. We will return to this topic in Chap. 7
- 31.
Metric spaces are considered in a topological context in Sect. 6.4.2
- 32.
How the basis tensors combine in this and other situations will be examined more closely in Chap. 7. Here, we lose nothing by focusing solely on the components.
- 33.
As stated in Sect. 1.2.2 “...wherever there is an equivalence relation, there is a quotient structure, and vice versa.”
- 34.
For sets, an identifying characteristic; for groups, an invariant subgroup; for rings, an invariant subring (ideal).
- 35.
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Starkovich, S.P. (2021). Vector and Tensor Spaces. In: The Structures of Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-73449-7_4
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