# Hierarchy of Mathematical Spaces

I really got lost among all those many different spaces in mathematics, and I got really confused what is special case of what.

For example, I knew for long time vector spaces, then Hilbert spaces, that I thought about as infinite dimensional vector spaces, then discovered that there is topological vector spaces, and that Hilbert spaces actually just a special case of, while the "usual" for me vector spaces are actually algebraetic vector spaces... etc.

Are there some visual, or written hierarchy model for those many different space? so one will understand what is the most fundamental object we start from? And what is a special case of what?

Something like Topology=>topological space=> if we equip it with metric we get.. => if we equip it with norm we get ..

Thank you in advance.

Edit

Something like to what is mentioned in Space (Wikipedia) or Connections between metrics, norms and scalar products.. (StackExchange) but with more details and subfields and what is the additional assumed structure.

## 1 Answer

This is a (probably icomplete) hierarchy of vector spaces from the point of view of functional analysis.

1. Vector spaces: algebraic structure (addition and multiplication by scalars)
2. Topological vector space: vector space with a topology such that addition and multiplication by scalars are continuous.
3. Locally convex topological vector spaces: TVS in which $0$ has a basis of convex neighbourhoods.
4. Fréchet space: LCTVS whose topology is derived from a translation invariant metric, complete.
5. Banach spaces: Fréchet space in which the metric is given by a norm.
6. Reflexive Banach spaces: the cannonical homomorphism between the space and its double dual is an isomorphism.
7. Hilbert space: Banach space in which the norm comes from an inner product.
• On the off chance it's not clear from context, Hilbert spaces are a type of reflexive Banach space. – Theo Bendit Jul 17 '15 at 11:58
• Every Hilbert space is a reflexive Banach space. – Julián Aguirre Jul 17 '15 at 15:49
• Ok good try but it makes me just more confused... for example: Fréchet space has metric, but LCTVS has not? then between them there should be Metric spaces I suppose? or Metric vector spaces? also somewhere between them I suppose should be also complete metric vector spaces also...etc, it is a mess! – TMS Jul 18 '15 at 11:50
• In all these spaces the algebraic structure and the topological structure are related. In TVS addition and multiplication by scalars must be continuous. Other topologies are, as far as I know, of little interest. In Fréchet spaces, the metric is translation invariant. And yes, there are non metrizable LCTVS. – Julián Aguirre Jul 18 '15 at 13:24