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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.

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    $\begingroup$ In Riesz's lemma, "the subspace is closed" means that it's a closed set when you view the whole space as a metric/topological space. This isn't intended to be a property of the subspace as a vector subspace, but rather just as a subset. It is definitely confusing that there's also the concept of being "closed under addition" and "closed under multiplying by scalars"! These are unrelated concepts, and I would expect a mathematician who's being careful, especially when writing about normed spaces, to always say "closed under ..." to refer to these concepts. $\endgroup$ Commented Oct 20, 2023 at 13:45

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1.A "closed" vector subspace is just a vector subspace that is closed topologically in some larger vector space. The property of addition and scalar multiplication being closed is part of the definition of a vector subspace, not the definition of a closed vector space.

2.Yes, all these concepts of "open" and "closed" are equivalent in the sense that the only way to define a topology on a space that has already got a metric is to use as basis elements all open balls in that space; and if the space has already got a topology, then all metrics (if any) must be compatible with that topology.

3.Yes, dimension certainly plays a role in topology. Actually, there is something called topological dimension, and it has interesting relationships with other definitions of dimension.

4.All norms on a finite-dimensional vector space over the same field are equivalent, thus a finite-dimensional normed space over the same field has a unique topology. Of course we could define other topologies on a finite-dimensional vector space, but they cannot be induced by a norm.

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