mynameiseva said:
A closed linear subspace is a subset of a vector space that is closed under vector addition and scalar multiplication. This means that if you take any two vectors in the subspace and add them together, the resulting vector will also be in the subspace. Similarly, if you multiply a vector in the subspace by a scalar, the resulting vector will also be in the subspace. In other words, a closed linear subspace is a subset of a vector space that is itself a vector space.
A closed linear subspace is a special type of subspace that satisfies the additional condition of being closed. This means that the subspace contains all of its limit points. In contrast, a general subspace may not be closed and may not contain all of its limit points. This distinction is important in the study of functional analysis and other areas of mathematics.
Examples of closed linear subspaces include the set of all polynomials of degree n or less, the set of all continuous functions on a closed interval, and the set of all square-integrable functions on a fixed interval. In general, any subset of a vector space that satisfies the closure condition can be considered a closed linear subspace.
The closure of a linear subspace is the smallest closed subspace that contains the subspace. In other words, it is the intersection of all closed subspaces that contain the subspace. The span of a subspace, on the other hand, is the set of all linear combinations of vectors in the subspace. While the span is always a subspace, it may not be closed. The closure of a subspace is always a closed subspace.
The concept of a closed linear subspace is important in mathematics because it allows us to study and analyze subsets of vector spaces that have additional properties. Closed linear subspaces are often used in functional analysis to describe spaces of functions that are complete in some sense. They also play a role in the study of topological vector spaces and other areas of mathematics.