Subspaces in Normed Spaces

Vector space on which a distance is defined.

In this unit, we will delve into the concept of subspaces in the context of normed spaces. A subspace is a subset of a vector space that is itself a vector space. In the context of normed spaces, we will explore how these subspaces behave and how they can be characterized.

Definition and Examples of Subspaces

A subspace of a normed space is a subset that is closed under vector addition and scalar multiplication. This means that if you take any two vectors in the subspace, their sum is also in the subspace. Similarly, if you take any vector in the subspace and multiply it by a scalar, the result is also in the subspace.

For example, consider the normed space of all real-valued continuous functions defined on the interval [0, 1], denoted by C([0, 1]). The set of all functions in C([0, 1]) that vanish at x = 0.5 is a subspace of C([0, 1]).

Linear Combinations and Span in Normed Spaces

A linear combination of vectors in a normed space is a vector that can be expressed as the sum of scalar multiples of the vectors. The span of a set of vectors in a normed space is the set of all possible linear combinations of the vectors. It is a subspace of the normed space.

Linear Independence and Basis in Normed Spaces

A set of vectors in a normed space is said to be linearly independent if no vector in the set can be expressed as a linear combination of the other vectors. A basis of a subspace is a set of linearly independent vectors that span the subspace. Every vector in the subspace can be expressed as a unique linear combination of the basis vectors.

Dimension of a Subspace in Normed Spaces

The dimension of a subspace is the number of vectors in any basis of the subspace. In finite-dimensional normed spaces, the dimension of any subspace is always finite. However, in infinite-dimensional normed spaces, subspaces may have infinite dimension.

In conclusion, understanding the concept of subspaces in normed spaces is crucial in the study of functional analysis and its applications. It provides a foundation for further study of more complex structures and concepts in normed spaces.