Self-Referential Definition of Orthogonality

There has for longer been an interest in finding equivalent conditions which define inner product spaces, and the respective literature is considerable, see for instance Amir, which lists 350 such results. Here, in this tradition, an alternative definition of orthogonality is presented which does not make use of any inner product. This definition, in the spirit of the recently developed non-wellfounded set theory, is self-referential, or circulatory. 0. Preliminaries Let us note that a group (G, ⋆) is defined by properties of its subgroups generated by no more than 3 elements. Indeed, the axioms of a group are (0.1) ∀ x, y ∈ G : x ⋆ y ∈ G (0.2) ∀ x, y, z ∈ G : x ⋆ (y ⋆ z) = (x ⋆ y) ⋆ z (0.3) ∃ e ∈ G : ∀ x ∈ G : x ⋆ e = e ⋆ x = x 1 (0.4) ∀ x ∈ G : ∃ x ′ ∈ G : x ⋆ x ′ = x ′ ⋆ x = e Clearly, vector spaces are also defined by properties of their vector subspaces generated by no more than 3 elements. On the other hand, the conditions that a vector space X be a normed space (X, || ||) will only involve its 2 dimensional vector subspaces, namely (0.5) ∀ x ∈ X : ||x|| ≥ 0 (0.6) ∀ x ∈ X : ||x|| = 0 ⇐⇒ x = 0 (0.7) ∀ x, y ∈ X : ||x+ y|| ≤ ||x||+ ||y|| Obviously, the same goes for finite dimensional Hilbert spaces, where the scalar product, and in particular, orthogonality only involve 2dimensional vector subspaces. In the case of infinite dimensional vector spaces, just like with infinite dimensional Banach spaces, completeness is also required, a condition which, of course, is no longer definable in terms of any finite dimensional vector subspaces. It can in general be recalled the important role of the structure of finite dimensional vector spaces in the study of arbitrary normed spaces, as illustrated by the earlier and more particular Khinchin, and the later Grothendieck inequalities. In the sequel, we shall consider the issue of orthogonality, and do so without the need to deal with completeness. 1. Linear Independence Let V be a vector space on R, and m ≥ 2, then we denote 2