Essentials from Vector Spaces

Let us start by recalling the definition of a vector space.

Definition 1.1.1. Consider a set V of elements called vectors and a field of scalars K. Given the summation rule + between vectors and the multiplication rule · between vectors and scalars, the tuple (V, K, +, · ) is called a vector space over the field K if V is closed under summation and multiplication by scalars, i.e., for every ~ψ, ~φ ∈ V and a, b ∈ K it holds that a ~ψ + b~φ ∈ V.

For our purpose, we can think of the field of scalars K to be complex numbers C. Moreover, on every vector space there exists a set of all transformation between vectors, an interesting subset of which are linear ones.

Definition 1.1.2. The set of all linear transformation on a vector space V (linear endomor-phisms) is formed from maps ˆA : V → V such that ˆA(a ~ψ + b~φ) = a ˆA( ~ψ) + b ˆA(~φ) ∈ V.

then (V, K, +, · , k · k) is called an normed linear vector space over the field K. Importantly, it is possible to define various norms on a vector space. As a particular case, if the field of scalars is real or complex numbers, then it is well-known that every inner product h · , · i naturally induces a norm, called `₂-norm, as

for every ~ψ ∈ V. As a consequence, every inner-product vector space is necessarily normed, but the converse is not true. We now recall the following.

Definition 1.1.3. The pair (S, d), where S is a set and d : S × S → R is a metric satisfying

It is well-known that for any normed vector space one can define the distance between two vectors ~ψ and ~φ as the norm of their difference vector, that is,

d( ~ψ, ~φ) =

Consequently, every normed vector space is indeed a metric space. Moreover, using Eq. (1.3), every inner-product space is also a metric space with respect to the distance function induced by the `₂-norm.

A very important notion in linear algebra is the completeness of vector spaces in the following sense. A complete vector space is one in which the limiting points of convergent sequences (Cauchy sequences) are included within the space. To put this in a rigorous form, we begin with the following.

Definition 1.1.4. Consider a sequence of vectors (e.g. vectors and points) S_seq = ( ~ψ_i)_i∈N in a normed space. S_seq converges to the limit ~ψ if and only if for every  > 0 there exists an integer N ∈ N such that for every n ∈ N and n > N it holds true that

Moreover, S_seq is called Cauchy if and only if for every  > 0 there exists an integer N ∈ N such that for every n, m ∈ N and n, m > N

The two notions of convergent above, despite their similarity, are different. For instance, the convergence of a sequence depends on the convergence point ~ψ which may or may not be within the space containing the sequence, whilst Cauchy convergence is a property of the sequence itself, regardless of the set containing the convergence point¹.

Lemma 1.1.1. A Cauchy sequence S_Cauchy = ( ~ψ_i)_i∈N on a set S is Cauchy on all supersets of S, as well as all subsets of S containing S_Cauchy.

There is, however, a necessary relationship between the two types of convergence.

Lemma 1.1.2. Every convergent sequence is Cauchy.

Thus, completeness is imposed by the Cauchy convergence as follows.

Definition 1.1.5. Consider the normed vector space V. Suppose that the sequence of vectors ( ~ψi)_i∈N ∈ V is Cauchy. Then, V is a complete vector space if it contains the converging points of all such Cauchy sequences.

A complete normed vector space is commonly called a Banach space. A particular case is obtained if the vector space is inner product and the norm is induced by the inner product as in Eq. (1.3). In this case, the resulting Banach space is called a Hilbert space. Hence, every Hilbert space is necessarily Banach, however, the converse does not hold true. It is important to bear in mind that all linear vector space can be completed by enlarging the space and adding the convergence point of all Cauchy sequences on them.

1The situation can be even worse, i.e., that the existence of the convergence point generally depends on the chosen notion of convergence.

Proposition 1.1.1. Any normed linear vector spaces can be completed to a Banach space.

Consequently, any inner-product linear vector space can be turned into a Hilbert space via completion.

Another very important notion in linear algebra is a linear functional.

Definition 1.1.6. The set of all linear functionals on V is formed by maps F : V → K from a vector space V to the scalar field K such that F(a ~ψ + b~φ) = aF(~ψ) + bF(~φ) ∈ K.

By defining the sum of two functionals F and F⁰ as (F + F⁰)( ~ψ) = F(~ψ) + F⁰( ~ψ), the set of all continuous linear functionals also becomes a linear vector space, called the dual space and denoted as V^d. For a Hilbert space H, the dual space H^d has a well-known structure given by Riesz theorem below.

Theorem 1.1.1. (Riesz Theorem.) Define the map M : V → V^d as Mψ~ = D ~ψ, · E

for every ψ ∈ H. The map~ M is an isomorphism between the vector space H and the dual vector space H^d.

The Riesz theorem 1.1.1 states that for every functional F ∈ H^d there exists an element ψ ∈ H such that~ F can be written uniquely as F = D ~ψ, ·E

. Consequently, by characterizing a complete inner product space H we have essentially characterized its dual space as well.

The notion of complete vector spaces in conjunction with Riesz theorem is widely used in the modern formulation of quantum mechanics in which the Dirac notation is exploited [1].

These two are the mathematical elements that bring rigorosity to the Dirac bra-ket notation.

Interestingly, the problem of incompleteness of inner-product vector spaces can arise only in infinite dimensional vector spaces.

Lemma 1.1.3. All norms on finite dimensional linear vector spaces are equivalent, in the sense that, given two norms

, there exists real positive numbers c₁ and c₂ such that

c₁

Proposition 1.1.2. All finite dimensional normed linear vector spaces are Banach spaces.

We close this section by the following well-known definitions and results.

Definition 1.1.7. A subset S_li = { ~ψ_i} is called linearly independent if no element of S_li can be written as a nontrivial linear combination of the rest of elements.

Definition 1.1.8. Suppose that the subset Sbasis ⊂ V of linearly independent vectors has the maximum cardinality among all linearly independent sets. Then, S_basis is called a basis set and d = cardS_basis is the dimensionality of the vector space.

Suppose that Sbasis = { ~ψi}ⁿ_i=1 is a basis vector for the n-dimensional vector space V. Hence, every element ~ψ ∈ V can be written as ~ψ =Pn

i=1x_iψ~_i with x_i ∈ K. We may ask how one can obtain the expansion coefficients x_i in this basis?

Definition 1.1.9. Given a basis set S_basis= { ~ψ_i}ⁿ_i=1 for a vector space V, the dual or reciprocal basis S_basis^d = {Fi}ⁿ_i=1 is a set of functionals in V^d such that Fi( ~ψ_j) = δ_ij for all i and j.

Using dual basis elements every expansion coefficient is thus given by x_i = Fi( ~ψ). In the particular case of a Hilbert space, using Riesz theorem 1.1.1, we have x_i =D ~φ^d_i, ~ψE

where the reciprocal basis is isomorphic to the set of vectors S_basis^d = {~φ^d_i}ⁿ_i=1⊂ H such thatD ~φ^d_i, ~ψ_jE

= δ_ij for all i and j.