Mathematics: A concept that is completely independent of real-world scenarios, and can be described correctly

There is an answer to the question [1]:

What is normed linear space, inner product space, metric space, Hilbert space? One of the characteristics of modern mathematics is to set as the object of study, the advantage is that many different problems can be abstracted from the essence of the same problem, of course, the disadvantage is to describe the more abstract, many people are difficult to understand.
Since it is a collection of studies, everyone is interested in different angles and the direction of research is different. In order to be able to study the set effectively, the set must be given some "structure" (a structure abstracted from some specific problems).
From the nature of mathematics, the most basic set has two types: linear space (a set of linear structures), metric space (a set of measurement structures).
For linear space, the main study of the description of the collection, intuitively speaking, is how to clearly tell the people what the collection is like. For the sake of clarity, the concept of a base (equivalent to a coordinate system in a three-dimensional space) is introduced, so that for a linear space, the elements in the collection can only know the coordinates of a given base if they know its base.
But the elements in the linear space have no "length" (equivalent to the length of the segments in the three-dimensional space), in order to quantify the elements in the linear space, a special "length", i.e. norm, is introduced into the linear space. The linear space assigned to the norm is called the linear space of the offender.
However, there is no concept of angle between two elements in normed linear space, in order to solve this problem, the concept of inner product is introduced in linear space.
Because there are measures, so we can introduce limits in metric space, normed linear space and inner product space, but there is a big difference between the limit of abstract space and the limit of real number is that the limit point may not be in the original given set, so the complete concept is introduced, and the complete inner product space is called Hilbert space.
The relationships between these spaces are:
Linear space and metric space are two different concepts with no intersection.
The normed linear space is the linear space of the norm and also the metric space (the metric space with linear structure).
Inner product space is normed linear space
Hilbert space is a complete inner product space.

In elementary mathematics, we define the arithmetic law; In modern mathematics, we introduce the concepts of sets and elements, define the concepts of space, vectors, inner product, norm, and so on----we may be puzzled as to why it can be applied to the actual analysis when it is completely divorced from the actual application scenario.

In fact, these mathematical concepts are not related to the actual application scenario, and the correlation between numbers and objects in the real world is artificially established. A number or matrix can refer to one or more entities, the number refers to which entity depends on the interpretation of the human, mathematically defined on the various concepts and operations of the numbers: space, vectors, inner product, norm, etc., they can correctly describe the actual application of the situation depends on the test of practice.

For example, numbers, vectors, matrices, and collections are mathematical objects that are often used by people to refer to geometric objects, such as points, line segments-----Note that the same number can also refer to different entities. After a simple correlation between the vector and the line segment, we magically discovered that it was possible to correctly determine whether the two segments were perpendicular by the mathematically defined inner product, that is, the event "two vectors within the product equals 0" and the event "two lines associated with the line perpendicular" always occurs at the same time or does not occur at the same time. Based on this conclusion, we establish a logical relationship between the inner product of the vector and the vertical relationship of the line segment, and our mathematical formula correctly describes the real scene.

[1] know: How to understand Hilbert space?

Mathematics: A concept that is completely independent of real-world scenarios, and can be described correctly