Nagasaka-po: Zhao Yun 7 in 7 out
Keep shoveling until the sky is full of color.
There was no increase; there was a buy-up trend.
The chart is for interested people to see, and the world is full of poetry.
Paying attention to “basic work” is a top priority.
1. Time only changes things that aren't strong enough in the first place.
2. Only those who have no new stories will remember the old ones.
3. People see benefits without harm; fish see food without being hooked.
4. Work hard for everything you want, and be relieved as soon as possible for everything you don't get.
5. As long as life is fun, that day's life is worth the ticket price.
6. Instead of striving for perfection and getting frustrated, it's better to persevere in a clumsy way.
2. Only those who have no new stories will remember the old ones.
3. People see benefits without harm; fish see food without being hooked.
4. Work hard for everything you want, and be relieved as soon as possible for everything you don't get.
5. As long as life is fun, that day's life is worth the ticket price.
6. Instead of striving for perfection and getting frustrated, it's better to persevere in a clumsy way.
Life needs four kinds of people
Famous teachers guide the way, noble people help each other, relatives support, villains stimulate. If you can integrate others, it means that you have the ability. If you are integrated by others, it means that you are valuable. No one will do it with the other. It is cooperation, mutual achievement, and mutual respect. Self-worth is very important, and character is a more precious value!
Famous teachers guide the way, noble people help each other, relatives support, villains stimulate. If you can integrate others, it means that you have the ability. If you are integrated by others, it means that you are valuable. No one will do it with the other. It is cooperation, mutual achievement, and mutual respect. Self-worth is very important, and character is a more precious value!
Functional analysis is a branch of modern mathematical analysis. It belongs to analytics. The main object of its study is a functional space composed of functions. The historical roots of functional analysis are the study of functional spaces and the study of the properties of function transformations. This view has proven to be particularly useful in the study of differential and integral equations. The use of the term functional function as an expression derives from variational law and represents a function acting on a function, which means that the parameter of a function is a function.
Functional analysis (English: Functional Analysis) is a branch of modern mathematical analysis. It belongs to analytics, and its main object of study is a functional space composed of functions. The historical roots of functional analysis are the study of functional spaces and the study of the properties of function transformations (such as Fourier transforms, etc.). This view has proven to be particularly useful in the study of differential and integral equations.
The use of the term functional function as an expression derives from variational law and represents a function acting on a function, which means that the parameter of a function is a function. This term was first used by Jacques Adama in a book on this subject in 1910. He is one of the main founders of functional analytical theory. However, the general concept of functional functions was previously introduced by Italian mathematician and physicist Vito Volterra (Vito Volterra) in 1887. Nonlinear functional theory continued to be studied by Jacques Adama's students, particularly Maurice Fréchet (Maurice Fréchet) and Levy (Levy).
Jacques Adama also founded the modern school of linear functional analysis, which was further developed by Frigyes Rees and the Lviv School of Mathematics (English: Lwów School of Mathematics), a group of Polish mathematicians around Stefan Banach (Stefan Banach).
From a modern point of view, functional analysis research is mainly a complete normalized linear space in the real or complex domain. This type of space is called Banach space, and the most important special case of Bach space is called Hilbert space, where the norm is derived from an internal product. This type of space is the basis for mathematical descriptions of quantum mechanics. More general functional analysis also studies spaces without defined norms, such as Fréchet spaces and topological vector spaces.
An important object of study in functional analysis is continuous linear operators in Banach space and Hilbert space. This type of operator can derive basic concepts of C*-algebra and other operator algebra.
The main theorems of functional analysis include:
The uniform bounded theorem (also known as the resonance theorem) describes the properties of a family of bounded operators.
The spectral theorem includes a series of results, the most commonly used of which gives an integral expression of a normal operator in Hilbert space. This result plays a central role in the mathematical description of quantum mechanics.
The Hahn-Banach Theorem (Hahn-Banach Theorem) studies how to extend an operator's norm from one subspace to the entire space. Another related result is the non-triviality of dual spaces.
Open mapping theorem and closed image theorem.
Most of the space studied by functional analysis is infinite-dimensional. To prove that an infinite-dimensional vector space has a set of bases, Zorn's Lemma (Zorn's Lemma) must be used. Furthermore, most important theorems in functional analysis are based on the Hahn-Banach theorem, and the theorem itself is a form where the axiom of choice (Axiom of Choice) is weaker than the Boolean prime ideal theorem (Boolean prime ideal theorem).
Functional analysis currently includes the following branches:
Soft analysis (soft analysis), the goal of which is to express mathematical analysis in the language of topological groups, topological rings, and topological vector spaces.
The geometry of the Banach space is represented by a series of works by Jean Bourgain.
Non-commutative geometry, the main contributors to this direction include Alain Connes, whose work is based in part on the results of George Mackey's theory of traversal.
Theories related to quantum mechanics are narrowly known as mathematical physics. From a broader perspective, as described by Israel Gellvand, they include most types of problems in representation theory.
Functional analysis (English: Functional Analysis) is a branch of modern mathematical analysis. It belongs to analytics, and its main object of study is a functional space composed of functions. The historical roots of functional analysis are the study of functional spaces and the study of the properties of function transformations (such as Fourier transforms, etc.). This view has proven to be particularly useful in the study of differential and integral equations.
The use of the term functional function as an expression derives from variational law and represents a function acting on a function, which means that the parameter of a function is a function. This term was first used by Jacques Adama in a book on this subject in 1910. He is one of the main founders of functional analytical theory. However, the general concept of functional functions was previously introduced by Italian mathematician and physicist Vito Volterra (Vito Volterra) in 1887. Nonlinear functional theory continued to be studied by Jacques Adama's students, particularly Maurice Fréchet (Maurice Fréchet) and Levy (Levy).
Jacques Adama also founded the modern school of linear functional analysis, which was further developed by Frigyes Rees and the Lviv School of Mathematics (English: Lwów School of Mathematics), a group of Polish mathematicians around Stefan Banach (Stefan Banach).
From a modern point of view, functional analysis research is mainly a complete normalized linear space in the real or complex domain. This type of space is called Banach space, and the most important special case of Bach space is called Hilbert space, where the norm is derived from an internal product. This type of space is the basis for mathematical descriptions of quantum mechanics. More general functional analysis also studies spaces without defined norms, such as Fréchet spaces and topological vector spaces.
An important object of study in functional analysis is continuous linear operators in Banach space and Hilbert space. This type of operator can derive basic concepts of C*-algebra and other operator algebra.
The main theorems of functional analysis include:
The uniform bounded theorem (also known as the resonance theorem) describes the properties of a family of bounded operators.
The spectral theorem includes a series of results, the most commonly used of which gives an integral expression of a normal operator in Hilbert space. This result plays a central role in the mathematical description of quantum mechanics.
The Hahn-Banach Theorem (Hahn-Banach Theorem) studies how to extend an operator's norm from one subspace to the entire space. Another related result is the non-triviality of dual spaces.
Open mapping theorem and closed image theorem.
Most of the space studied by functional analysis is infinite-dimensional. To prove that an infinite-dimensional vector space has a set of bases, Zorn's Lemma (Zorn's Lemma) must be used. Furthermore, most important theorems in functional analysis are based on the Hahn-Banach theorem, and the theorem itself is a form where the axiom of choice (Axiom of Choice) is weaker than the Boolean prime ideal theorem (Boolean prime ideal theorem).
Functional analysis currently includes the following branches:
Soft analysis (soft analysis), the goal of which is to express mathematical analysis in the language of topological groups, topological rings, and topological vector spaces.
The geometry of the Banach space is represented by a series of works by Jean Bourgain.
Non-commutative geometry, the main contributors to this direction include Alain Connes, whose work is based in part on the results of George Mackey's theory of traversal.
Theories related to quantum mechanics are narrowly known as mathematical physics. From a broader perspective, as described by Israel Gellvand, they include most types of problems in representation theory.
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成熟投资者：格局，概率，取舍。没有格局必然急功近利。不计概率会把运气当技术。不懂取舍，有所不为，最后必落入陷阱和圈套。
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