Function spaces

Definitions

Ring

A ring is a set where two operations, an "addition" and a "multiplication" can be defined.

Mathematical field/Körper/Cuerpo

a set is a field when an addition and multiplication can be defined (so, it's a ring but with additional restrictions) in such a way that both operations are associative, commutative and there are neuter and inverse elements. (Examples: ). Don't confuse this definition of a "field" with a scalar or vector field. In this document, is the set .

Function

a function f from a set A to a set B is a correspondence such that for each element there is exactly one corresponding element , . For example, A and B can be , or . Example of a function definition: .

In many programming languages, like in C or Fortran, you must specify the compiler what the domain of the function is. For example, the function is written in Fortran as

function f(x)
	real x
	complex f
	f = (0,1) + x
end function

We shall always try to specify the domain of the function to avoid mistakes.

Polynomial

a real polynomial is a function of the form . are the coefficients of the polynomial.

Vector space

a set is a vector space over a field if there is an addition and a scalar multiplication . The scalar multiplication should satisfy:

1. ;

2. ;

3. ;

4. , for all and .

Note: there is nothing in this definition that tell us what the elements of should be!

Note: if we also define a multiplication between the elements, we build the concept of an algebra.

Linear independence

The vectors are linearly independent when the equation only has the trivial solution . The maximum number of linearly independent vectors is called the dimension of the vector space.

Basis

there are linearly independent vectors for which each vector can be written as a linear combination of basis vectors: . There is always a bijective transformation between two given bases. The are the coordinates of the vector, usually written as a column vector (or transposed row vector to save space).

Example/Review: euclidean vector spaces

In previous courses, you have seen vector spaces defined over the field . A common basis are the vectors (Cartesian basis), which are linearly independent. A given vector is then expressed as

If we agree on a given basis, we can also express the vector as a n-tuple or column matrix:

It is common for the basis vectors to have a length of one. If this is the case, the length of a given vector is

Now imagine we generalise to n dimensions instead of the 3 we are used to. In , the distance between two vectors is given by

You have also encountered the inner product between two vectors in , which is defined as

Functions as vectors

The definition of a vector space doesn't tell us what the elements of the vector space are. Functions can also behave like vectors and form vector spaces.

Polynomials can form a vector space. Consider for example the polynomial . Its coordinates in the basis are , but in the basis , they are .

Exercise: Check that all the properties of a vector space are satisfied here.

What is the dimension of the vector space defined by polynomials? Let's take the definition of a polynomial. Given that a polynomial is for any , then the dimension is infinite. This is, in fact, a big difference between the vectors we have encountered so far and functions as vectors.

Another example of functions that build an infinitely-dimensional vector space is .

Normed spaces

Metric

It is the generalisation of the idea of the separation between two elements.

Norm

If we agree on where the "origin" of our space is located, then the norm gives us the "length" of the vector (like in Euclidean vectors). The definition of a norm is not unique. A norm must be non-negative and only vanish for the null element. The norm can also be used for defining the metric as

More formally, a norm over a real or complex vector field is a map such that it is

1. positive definite,
2. homogeneous, ;
3. it satisfies the triangle inequality.

There are many possible (equivalent) norms that can be defined in a vector space. Here we list some important ones:

• For , the norm generalises the idea of length such that . This is then called the Euclidean metric.
• For the complex numbers , the norm is .
• The Hilbert space of sequences is the set of real or complex sequences of vectors such that . This definition ensures that all the elements (sequences) are within the vector space.
• The function space over a subset of integrable functions, called , uses the norm . The function space defined with this metric is called the Hilbert space, and it is of great importance in Physics.

Inner product

We want to generalise the idea of the inner product between two vectors.

As a review: for Euclidean vectors in the field we had

For vectors in the field we have

For functions of the space defined in a given subset , we use the the norm to define the inner product

(The norm is then given by and the metric, ).

For this course, we use a more general definition of the inner product including a non-negative weigh function :

Just like with Euclidean vectors, we say that two vectors are orthogonal if

Many vectors (=functions) build orthogonal sets.

Last updated: 11 Aug 2025 at 12:27