Generalization of inner product spaces (analogue to uniform spaces/locally convex spaces)

Question

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.

---

Summary:
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

---

My questions:
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?

Answer 1

One can show that each (semi-)norm that satisfies the parallelogram identity

$$ 2\lVert x\rVert^2 + 2\lVert y\rVert^2 = \lVert x + y\rVert^2 + \lVert x - y\rVert^2 $$

is generated by a (semi-)inner product. This has been examined, e. g., in this question here on Math Stack Exchange. We can conclude:

• Inner product spaces are precisely those normed vector spaces whose norms satisfy the parallelogram identity.
• Semi-inner product spaces are precisely those semi-normed vector spaces whose semi-norms satisfy the parallelogram identity.
• ???-spaces are precisely those locally convex spaces of whose family of semi-norms each one satisfies the parallelogram identity.

According to a paper by Hicks and Huffman, Precupanu has named these spaces H-locally convex spaces. If such a space is complete, it is called a Generalized Hilbert space. Some properties:

• In the above paper, Hicks and Huffman prove some fixed point theorems - so, H-locally convex spaces seem to have some interesting properties that are/have been worth studying.
• Naimpally, Singh and Whitfield mention that nuclear spaces are a subset of generalized Hilbert spaces.

---

To give a summary, I reproduce the chart of spaces with H-locally convex spaces filled in:

POS. DEF. STRUCTURE   |   POS. SEMI-DEF. STRUCTURE   |  FAM. OF POS. SEMI-DEF. STRUCTURES
----------------------------------------------------------------------------------------
Inner product space  -->  Semi-inner product space  -->  H-locally convex spaces
        |                        |                          |
        V                        V                          V
Normed vector space  -->  Semi-normed vector space  -->  Locally convex space
        |                        |                          |
        V                        V                          V
Metric space         -->  Pseudometric space        -->  Uniform space

---