Author: qang pan
Link: https://www.zhihu.com/question/19967778/answer/28403912
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What are normed linear spaces, inner product spaces, metric spaces, and Hilbert spaces?
One of the characteristics of modern mathematics is that it takes sets as the research object. The advantage of this is that the essence of many different problems can be abstracted and turned into the same problem. Of course, the disadvantage of this is that the description is more abstract, and it is difficult for many people to understand. Since it is a research collection, each person is interested in different angles, and the research directions are also different. In order to be able to study collections effectively, collections must be given some "structure" (structure abstracted from some concrete problem). From the essence of mathematics, there are two basic types of sets: linear spaces (sets with linear structures) and metric spaces (sets with metric structures).
For linear space, the main study of the description of the set, intuitively is how to clearly tell others what the set looks like. In order to describe clearly, the concept of basis (equivalent to a coordinate system in three-dimensional space) is introduced, so for a linear space, as long as the basis is known, the elements in the set only need to know its coordinates under a given basis. Can. However, elements in linear space have no "length" (equivalent to the length of a line segment in three-dimensional space). In order to quantify elements in linear space, a special "length", that is, a norm, is introduced in linear space. A linear space given a norm is called a given linear space. But there is no concept of angle between two elements in normed linear space. To solve this problem, the concept of inner product is introduced into linear space. Because there are metrics, limits can be introduced in metric spaces, normed linear spaces, and inner product spaces, but the limit in abstract space is very different from the limit on real numbers. The limit point may not be in the original set. Therefore, the concept of completeness is introduced, and the complete inner product space is called Hilbert space. The relationship between these spaces is that linear space and metric space are two different concepts and have no intersection. A normed linear space is a linear space given a norm, and it is also a metric space (a metric space with a linear structure). The inner product space is a normed linear space. Hilbert space is a complete inner product space.