Finite Dimensional Normed Spaces and Subspaces

Vector space on which a distance is defined.

In the realm of mathematics, particularly in the study of functional analysis and related fields, finite dimensional normed spaces and their subspaces play a crucial role. This article aims to provide a comprehensive understanding of these concepts.

Finite Dimensional Normed Spaces

A normed space is a vector space on which a norm is defined. A norm is a function that assigns a strictly positive length or size to each vector in the vector space—save for the zero vector, which is assigned a length of zero. A finite dimensional normed space is a normed space with a finite basis.

The dimension of a normed space is the cardinality (number of vectors) in any basis. For a finite dimensional normed space, this is a finite number. The simplest example of a finite dimensional normed space is Euclidean space, which is a normed space of any finite dimension.

Properties of Finite Dimensional Normed Spaces

Finite dimensional normed spaces have several important properties:

1. Completeness: Every finite dimensional normed space is complete, meaning every Cauchy sequence in the space converges to a point in the space.

2. Compactness: In a finite dimensional normed space, a set is compact if and only if it is closed and bounded.

3. Convexity: Every finite dimensional normed space is convex, meaning for any two points in the space, the line segment that joins them is entirely contained within the space.

Subspaces in Finite Dimensional Normed Spaces

A subspace in a finite dimensional normed space is a subset of the space that is itself a normed space. The subspace must contain the zero vector, and it must be closed under vector addition and scalar multiplication.

Relationship Between a Normed Space and Its Subspaces

The relationship between a normed space and its subspaces is fundamental to understanding the structure of the space. Here are some key points:

1. Intersection of Subspaces: The intersection of any collection of subspaces of a normed space is also a subspace.

2. Sum of Subspaces: The sum of two subspaces is the smallest subspace that contains both.

3. Direct Sum: If the intersection of two subspaces is only the zero vector, the sum of the subspaces is called a direct sum.

Understanding finite dimensional normed spaces and their subspaces is crucial for further study in functional analysis, particularly in the study of Banach and Hilbert spaces. These concepts also have important applications in various fields such as quantum mechanics, signal processing, and differential equations.