normclosed

Normclosed is a term used in functional analysis and related fields of mathematics. It refers to a property of certain sets within a normed vector space. A set $C$ in a normed vector space $X$ is said to be normclosed if it contains all of its limit points with respect to the norm topology. In simpler terms, if you have a sequence of points within the set $C$ that converges to some point $x$, then that point $x$ must also be an element of $C$.

This property is equivalent to the set $C$ being a closed set in the standard topological sense, where the topology is induced by the norm of the vector space. The concept of a closed set is fundamental in topology and analysis. A set is closed if and only if its complement is open. For a normed vector space, the norm defines a natural distance, and consequently a natural topology.

Many important sets in functional analysis are normclosed. For instance, in a Banach space (a complete normed vector space), the closed unit ball is normclosed. Subspaces that are closed in the norm topology are also referred to as closed subspaces. The property of being normclosed is crucial for many theorems, such as the Hahn-Banach theorem and the open mapping theorem, which often require the domain or codomain to possess certain topological properties, including being closed.