When it comes to topologies, my understanding is that we designate elements of that topology as 'open.' In the context of a metric space, a topology is formed by selecting open sets, as defined with respect to a given metric. Moreover, a normed vector space is equipped with an induced metric derived from its norm. This metric allows us to associate it with a metric space and, consequently, with a topology.
In the realm of topology, a closed set is one for which the complement is open. In the context of metric spaces, it refers to the set containing all points to which sequences converge. Now, in regard to normed vector spaces, when discussing Banach spaces, it's often mentioned, as seen in Riesz's lemma, that a certain subspace is 'closed.' I wasn't familiar with a concrete definition of what a 'closed' vector space entails. To the best of my knowledge, this term has been used to describe the property of addition and scalar multiplication remaining closed operations.
Furthermore, are the concepts of 'open' and 'closed' equivalent in these structures? Does dimension play a role? In the case of finite dimensions, all norms are equivalent, and open sets are the same regardless of the induced metrics.
