📂Banach Space

Hamel Basis of Finite-Dimensional Vector Spaces

Definition ¹

Given a vector space $X$.

1. For vectors $x_{1} , \dots , x_{n}$ and scalar $\alpha_{1} , \dots , \alpha_{n}$ in $X$, $\alpha_{1} x_{1} + \cdots + \alpha_{n} x_{n}$ is called the linear combination of vectors $x_{1} , \dots , x_{n}$.

2. When it is $M =\left\{ x_{1} , \dots , x_{n} \right\}$, the set of all linear combinations of vectors of $M$ is called $\text{span} M$, which is a subspace of $X$ generated by $M$.

3. $M$ is said to be linearly independent if the only case satisfying $\alpha_{1} x_{1} + \cdots + \alpha_{n} x_{n} = 0$ is $\alpha_{1} = \cdots = \alpha_{n} = 0$.

4. If the finite set $K \subset X$ satisfies $\text{span} K = X$, then $X$ is said to be finite-dimensional.

5. When the linearly independent set $M$ satisfies $\text{span} M = X$, $M$ is called a basis of $X$.

6. The cardinality $\dim X := | M|$ of the basis is called the dimension of $X$.

Description

In vector spaces, the basis is especially referred to as the Hamel basis when discussing ‘finite’ linear combinations. A finite-dimensional normed space might sound complex, but it’s a familiar concept from the beginning of linear algebra studies. Usually, the interesting properties of these spaces are taken for granted without much serious contemplation.

• A finite-dimensional normed space has a basis.
• All norms defined on a finite-dimensional vector space are equivalent.
• A finite-dimensional normed space is complete.

Although these facts are taken as given when considering Euclidean spaces, the same cannot be said for general spaces. Each statement requires proof, and the process is not always straightforward.

See Also

• Linear independence, basis, and dimension in linear algebra
• Basis of infinite-dimensional vector spaces: Schauder basis