In the following I am going to devise a chart of topological spaces that contains inner product spaces, normed vector spaces, metric spaces and other related spaces. In the end there will be a gap in the chart and, essentially, my question is how to fill this gap.
Step 1: Spaces with positive definite structures
A topological space can be generated, e. g., by some structure like inner products, norms and metrics. Inner products, norms and metrics have one property in common: They are positive definite.
- $\langle x,x\rangle \geq 0$ and $(\langle x,x\rangle =0 \implies x=0)$
- $||x|| \geq 0$ and $(||x||=0 \implies x=0)$
- $d(x,y) \geq 0$ and $(d(x,y)=0 \implies x=y)$
It is well known that each inner product induces a norm and each norm induces a metric.
Step 2: Spaces with positive semi-definite structures
If we relax the positive definite properties a bit, we are left with positive semi-definite structures:
- Instead of inner products we then have symmetric positive semi-definite bilinear forms (or hermitian positive semi-definite sesquilinear forms). I'll call them semi-inner products below; please note that this name isn't used consistently in literature/internet.
- Instead of norms we then have semi-norms.
- Instead of metrics we then have pseudometrics.
And again, one can show that each semi-inner product induces a semi-norm and each semi-norm induces a pseudometric.
Step 3: Spaces with families of positive semi-definite structures
- The generalization of a metric space is called a uniform space. Uniform structures are connected to pseudometrics: one way to define uniform spaces is via a family of pseudometrics.
See for example Bourbaki, General Topology II, Chapter IX, §1, section 2, Definition of a uniformity by means of a family of pseudometrics.
- The generalization of a normed vector space is called a locally convex space. Locally convex spaces can be defined by a family of semi-norms on a vector space.
See for example the Wikipedia article on locally convex spaces.
Locally convex spaces can be interpreted as a subclass of uniform spaces.
Essentially, what we get is the following chart of structures and spaces, where every arrow can be read as "is a/can be interpreted as a":
POS. DEF. STRUCTURE | POS. SEMI-DEF. STRUCTURE | FAM. OF POS. SEMI-DEF. STRUCTURES ---------------------------------------------------------------------------------------- Inner product space --> Semi-inner product space --> ??? | | | V V V Normed vector space --> Semi-normed vector space --> Locally convex space | | | V V V Metric space --> Pseudometric space --> Uniform space
How can the gap ("???") in the chart above be filled? Is there a generalization of an inner product space that is constructed using a family of semi-inner products similarly to how uniform spaces and locally convex spaces can be constructed by families of pseudometrics and semi-norms? Do these generalized inner product spaces have a special name I haven't found out yet and have they ever been systematically studied?