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2.1 Definition of a linear vector space

A linear vector space V over a scalar set S (we shall typically consider sets S = ℝ or ℂ) is a set of objects (called vectors) a, b, c, \mathop{\mathop{…}} with two operations:

  1. Addition of any two vectors, c = a + b;
  2. Multiplication by a scalar λ ∈ S, b = λa.

These must satisfy the following conditions

  1. V is closed under addition, \mathop{∀}a,b ∈ V : a + b ∈ V .
  2. Addition is commutative:
    \mathop{∀}a,b ∈ V : a + b = b + a

    and associative

    \mathop{∀}a,b,c ∈ V : (a + b) + c = a + (b + c).

  3. There exists a null vector 0 ∈ V , \mathop{∀}a ∈ V : a + 0 = a.
  4. Every element a ∈ V has an inverse −a ∈ V such that a + (−a) = 0.
  5. The set V is closed under multiplication by a scalar, \mathop{∀}a ∈ V,λ ∈ S : λa ∈ V .
  6. The multiplication is distributive for addition of both vectors and scalars,
    \eqalignno{ \mathop{∀}a,b ∈ V,λ ∈ S : \quad &λ(a + b) = λa + λb, & & \cr \mathop{∀}a ∈ V,λ,μ ∈ S : \quad &(λ + μ)a = λa + μa, & & }

    and associative,

    \eqalignno{ \mathop{∀}a ∈ V,λ,μ ∈ S : &λ(μa) = (λμ)a. & & }
  7. There is a unit element 1 in S, such that 1a = a.

Note that we have not defined subtraction; it is derived operation, and is defined through the addition of an inverse element.

Example 2.1: 

The space {ℝ}^{3} of vectors r = \left (\array{ x\cr y \cr z} \right ) = xi+yj+zk is a vector space over the set S = ℝ.

Example 2.2: 

The space of two-dimensional complex spinors

\left (\array{ α\cr β } \right ) = α\left (\array{ 1\cr 0} \right )+β\left (\array{ 0\cr 1} \right ),

α,β ∈ ℂ, is a vector space.

Note: If we look at the space of up and down spins, we must require that the length of the vectors (the probability), |α{|}^{2} + |β{|}^{2}, is 1. This is not a vector space, since

{ \left |\left (\array{ {α}_{1} \cr {β}_{1} } \right ) + \left (\array{ {α}_{2} \cr {β}_{2} } \right )\right |}^{2} = |{α}_{ 1}+{α}_{2}{|}^{2}+|{β}_{ 1}+{β}_{2}{|}^{2} = |{α}_{ 1}{|}^{2}+|{β}_{ 1}{|}^{2}+|{α}_{ 2}{|}^{2}+|{β}_{ 2}{|}^{2}+2ℜ({α}_{ 1}^{∗}{α}_{ 2}+{β}_{1}^{∗}{β}_{ 2}),

which is not necessarily equal to 1.

Example 2.3: 

The space of all square integrable (i.e., all functions f with \mathop{\mathop{\mathop{∫ }\nolimits }}\nolimits dx\kern 1.66702pt |f(x){|}^{2} < ∞) complex functions f of a real variable, f : ℝ\mathrel{↦}ℂ is a vector space, for S = ℂ.

The space defined above is of crucial importance in Quantum Mechanics. These wave functions are normalisable (i.e., we can define one with total probability 1).

The space of all functions f, f : ℝ\mathrel{↦}ℂ with \mathop{\mathop{\mathop{∫ }\nolimits }}\nolimits dx\kern 1.66702pt |f(x){|}^{2} < ∞ is denoted as {ℒ}^{2}(ℝ).

2.1.1 Problems

1.

Show that the zero vector 0 is unique, and that for each a there is only one inverse −a.

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