Cuchulainn wrote:*The construct is one of the Euclidean space generalisations - the Hilbert space. The fact that it contains all such possible states of the system stems from its completeness, and the completeness relies on the Cauchy convergence.*

Good remark. In Hilbert spaces H you can define an inner product and then a norm induced by that inner products. All kinds of norms can be defined. So, a Cauchy sequence will converge to an element _in_ H,

There are spaces that are not complete.

Some spaces only have a metric and then induced norm. Banach spaces are complete metric spaces. These with Sobolev spaces) are the heavy-duty spaces for applied functional analysis and standard fodder for PDE/FEM. This was my favourite stuff as undergrad.

Hilbert spaces are rare beasts in the infested jungle of Banach spaces.. The parallelogram law can break down for some spaces and hence is not an inner product space).

Thinking out loud: norms for ML, Hilbert, Banach, complete etc etc.

The question of whether something is real or in the imagination is not a well posed one. It is impossible to distinguish the two. Even what you consider as seeing is the mental construct from the photons hitting your retina having already filtered by the retina and its photosensors and neurons not to mention the layers of neural networks in the brain. As Einstein put it well "what you observe is determined by your theory."

Is an integer real or imaginary, is a fraction real or imaginary, is a real number, is a complex number? Consider the present example of a complete space.

Banach space is a complete normed space. However, a norm is not necessary for the characterization of completeness. So long the space is a topological vector space, a Cauchy sequence can be defined. Whether a topological vector space is complete or not, is only a technicality, rather than an essential feature of a space. For an incomplete topological space, we can construct a new topological vector space by defining all Cauchy sequences of the original space as elements of a new space. The new topology consists of the closed sets each of which is the union of an original open set and all of its Cauchy sequences. We preserve all the binary operators like addition and multiplication. This newly constructed vector space is complete and preserves all the previous operations.

In fact, completion is used wider than just the topological vector space extension. We complete a set in many myriad ways just so that an operation can be carried out. The very notion of fraction comes from the completion of the division of integers, the negative numbers from the completion of subtraction, the real transcendental number from the completion of solving integral polynomial equations, the irrational number from the Cauchy sequence on the real line, and the complex number from making up the roots of real polynomial equations. These are all mental constructs. Are they real or imaginary?