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In real Euclidean spaces ℝ² and ℝ³, the inner product of two vectors v, w is defined as the real number:

where ϑ is the smallest angle between v and w and ‖ ‖ represents the norm (or the magnitude) of the vectors.

Using the inner product, it is possible to define the orthogonal projection of vector v in the direction defined by vector w. A distinction must be made between:

1 – the scalar projection of v in the direction of ; and

2 – the vector projection of v in the direction of ;

where is the unit vector in the direction of w. Evidently, the roles of v and w can be reversed.

The absolute value of the scalar projection measures the “similarity” of the directions of two vectors. To understand this concept, consider two remarkable relative positions between v and w:

1 – if v and w possess the same direction, then the angle between them ϑ is either null or π, hence cos(ϑ) = ±1, that is, the absolute value of the scalar projection of v in direction w is ‖v‖;

2 – however, if v and w are perpendicular, then and hence cos(ϑ) = 0, showing that the scalar projection of v in direction w is null.

When the position of v relative to w falls somewhere in the interval between the two vectors described above, the absolute value of the scalar projection of v in the direction of w falls between 0 and ‖v‖; this explains its use to measure the similarity of the direction of vectors.

In this book, we shall consider vector spaces which are far more complex than ℝ² and ℝ³, and the measure of vector similarity obtained through projection supplies crucial information concerning the coherence of directions.

Before we can obtain this information, we must begin by moving from Euclidean spaces ℝ² and ℝ³ to abstract vector spaces. The general definition of an inner product and an orthogonal projection in these spaces may be seen as an extension of the previous definitions, permitting their application to spaces in which our representation of vectors is no longer applicable.

Geometric properties, which can only be apprehended and, notably, visualized in two or three dimensions, must be replaced by a set of algebraic properties which can be used in any dimension.

Evidently, these algebraic properties must be necessary and sufficient to characterize the inner product of vectors in a plane or in real space. This approach, in which we generalize concepts which are “intuitive” in two or three dimensions, is a classic approach in mathematics.

In this chapter, the symbol V will be used to describe a vector space defined over the field , where is either ℝ or and is of finite dimension n < +∞. field contains the scalars used to construct linear combinations between vectors in V . Note that two finite dimensional vector spaces are isomorphic if and only if they are of the same dimension. Furthermore, if we establish a basis B = (b₁, . . . , b_n) for V , an isomorphism between V and ⁿ can be constructed as follows:

that is, I associates each v ∈ V with the vector of ⁿ given by the scalar components of v in relation to the established basis B. Since I is an isomorphism, it follows that ⁿ is the prototype of all vector spaces of dimension n over a field .

DEFINITION 1.1.– Let V be a vector space defined over a field .

A -form over V is an application defined over V × V with values in , that is:

DEFINITION 1.2.– Let V be a real vector space. A couple (V, 〈, 〉) is said to be a real inner product space (or a real pre-Hilbert space) if the form 〈, 〉 is:

1) bilinear, i.e.1 linear in relation to each argument (the other being fixed):

and:

2) symmetrical: 〈v, w〉 = 〈w, v〉, ∀v, w ∈ V ;

3) defined: 〈v, v〉 = 0 v = 0_V , the null vector of the vector space V ;

4) positive: 〈v, v〉 > 0 ∀v ∈ V , v ≠ 0_V .

Upon reflection, we see that, for a real form over V , the symmetry and bilinearity requirements are equivalent to requiring symmetry and linearity on the left-hand side, that is:

The simplest and most important example of a real inner product is the canonical inner product, defined as follows: let v = (v₁, v₂, . . . , v_n), w = (w₁, w₂, . . . , w_n) be two vectors in ℝⁿ written with their components in relation to any given, but fixed, basis in ℝⁿ. The canonical inner product of v and w is:

where v^t and w^t in the final equations are the transposed vectors of v and w, giving us the matrix product of a line vector (treated as a 1 × n matrix) and a column vector (treated as an n × 1 matrix).

The extension of these definitions to complex vector spaces is not particularly straightforward. First, note that if V is a complex vector space, then there is no bilinear and definite-positive transformation over V × V . In this case, any vector v ∈ V would give the following:

As we shall see, the property of positivity is essential in order to define a norm (and thus a distance, and by extension, a topology) from a complex inner product. To obtain an algebraic structure for complex scalar products which remains compatible with a topological structure, we are therefore forced to abandon the notion of bilinearity, and to search for an alternative.

We could consider antilinearity², i.e.

But it has the same problem as bilinearity, 〈iv, iv〉 = (−i)(−i)〈v, v〉 = i²〈v, v〉 = −〈v, v〉² ≼ 0.

A simple analysis shows that, in order to avoid losing the positivity, it is sufficient to request the linearity with respect to one variable and the antilinearity with respect to the other. This property is called sesquilinearity³.

The choice of the linear and antilinear variable is entirely arbitrary.

By convention, the antilinear component is placed on the right-hand side in mathematics, but on the left-hand side in physics.

We have chosen to adopt the mathematical convention here, i.e. 〈αv, βw〉 = αβ̅〈v, w〉.

Next, it is important to note that sesquilinearity and symmetry are incompatible: if both properties were verified, then 〈v, αw〉 = 〈v, w〉, and also 〈v, αw〉 = 〈αw, v〉 = α〈w, v〉 = α〈v, w〉. Thus, 〈v, αw〉 = 〈v, w〉 = α〈v, w〉 which can only be verified if α ∈ ℝ.

Thus 〈, 〉 cannot be both sesquilinear and symmetrical when working with vectors belonging to a complex vector space.

The example shown above demonstrates that, instead of symmetry, the property which must be verified for every vector pair v, w is , that is, changing the order of the vectors in 〈, 〉 must be equivalent to complex conjugation.

A transform which verifies this property is said to be Hermitian⁴.

These observations provide full justification for Definition 1.3.

DEFINITION 1.3.– Let V be a complex vector space. The pair (V, 〈, 〉) is said to be a complex inner product space (or a complex pre-Hilbert space) if 〈, 〉 is a complex form which is:

1) sesquilinear:

∀ v₁, v₂, w₁, w₂ ∈ V , and:

∀ α, β ∈ , ∀ v, w ∈ V ;

2) Hermitian: , ∀v, w ∈ V ;

3) definite: 〈v, v〉 = 0 v = 0_V , the null vector of the vector space V;

4) positive: 〈v, v〉 > 0 ∀v ∈ V , v ≠ 0_V .

As in the case of the canonical inner product, for a complex form over V , the symmetry and sesquilinearity requirement is equivalent to requiring the Hermitian property and linearity on the left-hand side; if these properties are verified, then:

Considering the sum of n, rather than two, vectors, sesquilinearity is represented by the following formulae:

[1.1]

[1.2]

In ⁿ, the complex Euclidean inner product is defined by:

where v = (v₁, v₂, . . . , v_n), w = (w₁, w₂, . . . , w_n) ∈ ⁿ are written with their components in relation to any given, but fixed, basis in ⁿ.

The symbol will be used throughout to represent either ℝ or in the context of properties which are valid independently of the reality or complexity of the inner product.

THEOREM 1.1.– Let (V, 〈 , 〉) be an inner product space. We have:

1) 〈v, 0_V 〉 = 0 ∀v ∈ V ;

2) if 〈u, w〉 = 〈v, w〉 ∀w ∈ V , then u and v must coincide;

3) 〈v, w〉 = 0 ∀v ∈ V w = 0_V , i.e. the null vector is the only vector which is orthogonal to all of the other vectors.

PROOF.–

1) 〈v, 0_V 〉 = 〈v, 0_V + 0_V 〉 = 〈v, 0_V 〉 + 〈v, 0_V 〉 by linearity, i.e. 〈v, 0_V 〉 − 〈v, 0_V 〉 = 0 = 〈v, 0_V 〉.

2) 〈u, w〉 = 〈v, w〉 ∀w ∈ V implies, by linearity, that 〈u − v, w〉 = 0 ∀w ∈ V and thus, notably, considering w = u − v, we obtain 〈u − v, u − v〉 = 0, implying, due to the definite positiveness of the inner product, that u − v = 0_V , i.e. u = v.

3) If w = 0_V , then 〈v, w〉 = 0 ∀v ∈ V using property (1). Inversely, by hypothesis, it holds that 〈v, w〉 = 0 = 〈v, 0_V 〉 ∀v ∈ V , but then property (2) implies that w = 0_V .

Finally, let us consider a typical property of the complex inner product, which results directly from a property of complex numbers.

THEOREM 1.2.– Let (V, 〈 , 〉) be a complex inner product space. Thus:

PROOF.– Consider any complex number z = a + ib, so −iz = b − ia, hence b = ℑ (z) = ℜ (−iz). Taking z = 〈v, w〉, we obtain ℑ (〈v, w〉) = ℜ (−i〈v, w〉) = ℜ (〈v, iw〉) by sesquilinearity.