notes/docs/mathematics/functional-analysis/normed-spaces/vector-spaces.md

Vector spaces

Definition 1: a vector space X over a scalar field F is a non-empty set, on which two algebraic operations are defined; vector addition and scalar multiplication. Such that

1. (X, +) is a commutative group with neutral element 0.
2. the scalar multiplication satisfies \forall x, y \in X and \lambda, \mu \in F
   • \lambda (x + y) = \lambda x + \lambda y,
   • (\lambda + \mu) x = \lambda x + \mu x,
   • \lambda (\mu x) = (\lambda \mu) x,
   • 1 x = x.

When F = \mathbb{R} we have a real vector space while when F = \mathbb{C} we have a complex vector space.

We have that the metric spaces \mathbb{R}^n, C, l^p and l^\infty are also vector spaces.

??? note "Proof:"

I am too lazy to add this trivial proof. Maybe some time in the future, if I do not forget.

Definition 2: a subspace of a vector space X is a non-empty subset M of X, such that \forall x, y \in M and \lambda, \mu \in F:

\lambda x + \mu y \in M,

with M itself a vector space.

A special subspace M of a vector space X is the improper subspace M = X. Every other subspace of X is a proper subspace.

Linear combinations

Definition 3: a linear combination of the vectors \{x_i\}_{i=1}^n with n \in \mathbb{N} is vector of the form

\alpha_1 x_1 + \dots + \alpha_n x_n = \sum_{i=1}^n \alpha_i x_i,

with \{\alpha_i\}_{i=1}^n \in F.

The set of all linear combinations of a set of vectors is defined as follows.

Definition 4: the span of a subset M \subset X of a vector space X, denoted by \mathrm{span}(M), is the set of all linear combinations of vectors from M.

It follows that \mathrm{span}(M) is a subspace of X.

Linear independence

Definition 5: a finite subset of vectors M = \{x_i\}_{i=1}^n is linearly independent if

\sum_{i=1}^n \alpha_i x_i = 0 \implies \forall i \in {1, \dots, n}: \alpha_i = 0.

The converse may also be defined.

Definition 6: a finite subset of vectors M = \{x_i\}_{i=1}^n is linearly dependent if \exists \{\alpha_i\}_{i=1}^n \in F not all zero such that

\sum_{i=1}^n \alpha_i x_i = 0.

The notions of linear dependence and independence may also be extended to infinite subsets.

Definition 7: a subset M of a vector space X is linearly independent if every non-empty finite subset of M is linearly independent.

While the converse in this case is defined by the contradiction.

Definition 8: a subset M of a vector space X is linearly dependent if M is not linearly independent.

Dimension and basis

Definition 9: a vector space X is finite dimensional if there exists a n \in \mathbb{N}, such that X contains a set of n linearly independent vectors, while every set of n+1 vectors in X is linearly dependent. In this case n is the dimension of X, denoted by \dim X = n.

By definition X = \{0\} is finite dimensional and \dim X = 0.

Definition 10: if a vector space X is not finite dimensional then X is infinite dimensional.

The following definition of a basis is both relevant to finite and infinite dimensional vector spaces.

Definition 11: a basis B of a vector space X is a linearly independent subset of X, that spans X.

Such a set B is also called a Hamel basis of X.

Theorem 1: every vector space X has a Hamel basis.

??? note "Proof:"

Read it again, a proof is not necessary.

Theorem 2: let X be a vector space with \dim X = n \in \mathbb{N}. Then any proper subspace M \subset X has dimension less than n.

??? note "Proof:"

If $n = 0$, then $X = \{0\}$ and $X$ has no proper subspace.

If $\dim M = 0$, then $M = \{0\}$ and $X \neq M \implies \dim X \geq 1$. 

If $\dim M = n$ then $M$ would have a basis of $n$ elements, which would also be a basis for $X$ since $\dim X = n$, so that $X = M$. 

This shows that any linearly independent set of vectors in $M$ must have fewer than $n$ elements and $\dim M < n$.