From Euclidean to Hilbert Spaces

1Inner Product Spaces (Pre-Hilbert)

This chapter will focus on inner product spaces, that is, vector spaces with a scalar product, specifically those of finite dimension.

1.1. Real and complex inner products

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, ...