Hilbert space (Hilbert space)

Reference:
https://www.zhihu.com/question/19967778
http://blog.csdn.net/mr_hai_cn/article/details/53207307

Hilbert space (Hilbert space)

In Hilbert space, the basis is generally a function. The common plane wave function contains various frequencies. One frequency corresponds to a basis, and the dimension is infinite. These bases, that is, the plane wave function are complete (any element in the Hilbert space can be expanded by the plane wave function, which actually refers to the Fourier transform), and orthogonal (the plane wave function is "dot product" as a delta function).
In other words, Hilbert space is inner product space + completeness .
So what is inner product space? An inner product is defined on a linear space, which is called an inner product space. Inner product can form in space such as intersection angle, vertical and projection, etc., so it is customary to call it Euclidean space. Therefore, the space we live in on weekdays is Euclidean space.
The completeness simply means that the space is in the limit operation, and the limit cannot be run out. Therefore, it is obvious that the set of rational numbers and the set of irrational numbers are not complete. The set of real numbers is complete.

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