separation theorem

Statement of Separation Theorem

The Separation Theorem is a mathematical theorem used in economics and other fields in order to describe the possibility of separating a problem into two independent components. In economics, it is basically used to explain how a market or economy works. The theorem states that a market equilibrium can be achieved through the separate decision-making of different economic agents.

The Separation Theorem became important in economics after the publication of Kenneth Arrow’s paper “The Theory of Price” in 1951. In the paper, Arrow established that a competitive market could internalize some of the externalities. This means that the individual decisions of market participants could lead to a social-optimal outcome. This was a major discovery in economics, changing the way the field was viewed.

The Separation Theorem is based on a few basic principles:

1. There is perfect competition in the market which means that there are many different potential buyers and sellers and no single agent has any undue power or influence over the market.

2. Market participants are rational, meaning they make decisions based on their own preferences and desires.

3. Prices are determined by how much demand there is relative to the amount of supply in the market.

4. Markets are self-regulating, meaning if the price is too high, more supply will enter the market and if the price is too low, demand will decrease.

In order to understand the implications of the Separation Theorem, it is important to understand the concept of “price equilibrium”. This is the point at which the market comes to a balance between buyers and sellers. At this point, the prices of goods determine the demand and supply, and no single buyer or seller can have an undue influence on the price. The Separation Theorem claims that in equilibrium, the decision that buyers and sellers make are independent of each other, and no single buyer or seller can have an influence on the entire market.

The Separation Theorem is essential for understanding how markets and the economy work. The theorem tells us that the decisions made by individuals have an effect on the entire market. By having a better understanding of the underlying market forces, economists are able to make better predictions about how the economy works and what future economic trends might be.

The Separation Theorem is just one of the many theories and models developed by economists to explain how the economy works. In addition to this, the Theory of Games is another important concept developed by economists to understand how markets function. While the two theories are interrelated, they are not quite the same.

The Separation Theorem is an important and useful concept in economics. It demonstrates how a market can be broken down into separate decision-makers who each have an impact on the market equilibrium. The theorem explains the self-regulating nature of markets and how the decisions taken by buyers and sellers are independent and have a direct effect on the overall price level of goods.

Overall, the Separation Theorem is a very useful tool for economists to understand the behavior of markets and the economy as a whole. It should be used in conjunction with other economic theories to get a better understanding of the inner workings of the economy.

The Banach Separation Theorem is a mathematical theorem which states that if two non-empty convex sets in a Banach space are separated by a hyperplane, (in other words the two sets are in two different open half spaces determined by the hyperplane) then it follows that there is a non-zero vector in the Banach space, bounded to the hyperplane, such that the inner product of the vector and any given point in the first set is lower than the inner product of the vector and any given point in the second set.

In other words, the Banach Separation Theorem states that for two convex sets in a Banach space, if the sets can be separated by a separating hyperplane, then there exists a nonzero vector in the Banach space which is proportional to the separating hyperplane and whose inner product with the points in the first set is less than its inner produc with the points in the second set.

The Banach Separation Theorem is closely related to the Hahn-Banach theorem, a theorem from functional analysis, which states that any linear functional on a convex subset of a Banach space can be extended to an affine functional on the entire space. The connection between the two theorems is that the Hahn-Banach theorem is used to construct the separating hyperplane, while the Banach Separation Theorem provides a vector in the direction of the separating hyperplane which satisfies the required property.

The Banach Separation Theorem has many applications in the fields of optimization, game theory and economics. It is used in the Krein-Milman theorem, a theorem in convex analysis which states that any non-empty convex set in a locally convex topological vector space is the closed convex hull of its extreme points. It is also used in the minimax theorem, which is an application of the Banach-Mazur theorem.

In conclusion, the Banach Separation Theorem is a fundamental tool in the study of convex sets in Banach spaces. It has applications in various fields, from economics and optimization to game theory and convex analysis.