Hahn Banach theorem example

In summary, the Hahn Banach theorem states that any bounded linear functional on a subspace of a normed linear space can be extended to a linear functional on the entire space with the same norm. The proof of this theorem is non-constructive and uses Zorn's lemma. To better understand this theorem, an example is given where the functional on a subspace is defined by taking the limit of a sequence. It is then shown that this functional can be extended to the entire space. However, it is difficult to imagine what this extended functional would look like. It would have to satisfy certain properties, such as giving the same value for sequences with converging differences and having the sum of the values for two sequences equal the limit
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I'm trying to understand the Hahn Banach theorem, that every bounded linear functional f on some subspace M of a normed linear space X can be extended to a linear functional F on all of X with the same norm, and which agrees with f on M. But the proof is non-constructive, using zorn's lemma.

So I'm trying to come up with examples so I can understand it better. For example, let L be the space of all bounded infinite sequences with the sup norm, and let M be the subspace of L consisting of those sequences that converge to some finite limit. Then, on this subspace, the functional given by taking the limit of a sequence is clearly linear and bounded. So it must extend to a bounded linear functional on all of L. But I can't imagine what such a functional would look like. Is it possible to explicitly construct one?
 
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The functional would have to have the following properties:

1. Any two sequences whose difference converges to zero must get the same value (eg, if f(1,0,1,0,...)=a then f(1/2,1/2,2/3,1/3,3/4,1/4,...)=a).

2. If the sum of two sequence converges to some limit, the sum of the values for each sequence equals the limit of their sum. (eg, if f(1,0,1,0,...)=a, then f(0,1,0,1,...)=1-a).

But what is, say, a? Can it be anything?
 
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First of all, it is great that you are trying to come up with examples to better understand the Hahn Banach theorem. It is a fundamental result in functional analysis and understanding it through examples can definitely help solidify your understanding.

In your example, you have correctly identified a subspace M of a normed linear space X and a bounded linear functional f on M. The Hahn Banach theorem guarantees the existence of a linear functional F on all of X that extends f and has the same norm. However, as you mentioned, the proof of this theorem is non-constructive and relies on the use of Zorn's lemma.

Unfortunately, it is not possible to explicitly construct such a functional F in this case. This is because the existence of such a functional relies on the axiom of choice, which is a fundamental principle in set theory. It states that for any collection of non-empty sets, it is possible to choose one element from each set. In the context of the Hahn Banach theorem, this means that we can choose a linear functional F that extends f on M, but we cannot explicitly construct it.

To better understand this, let's look at a simpler example. Consider the space of real numbers R and the subspace of rational numbers Q. We know that every real number can be approximated by a rational number, but we cannot explicitly construct a rational number that approximates a given real number. This is because the existence of such a rational number relies on the axiom of choice.

In the same way, the existence of a linear functional F that extends f on M in your example relies on the axiom of choice, and hence, cannot be explicitly constructed. However, we can still understand the properties of this functional F through the Hahn Banach theorem and its proof.

In conclusion, while it may be difficult to explicitly construct a functional F that extends f on M in your example, understanding the Hahn Banach theorem and its proof can help us understand its properties and significance in functional analysis.
 

Related to Hahn Banach theorem example

What is the Hahn Banach theorem example?

The Hahn Banach theorem is a fundamental result in functional analysis that deals with the extension of functionals defined on a subspace to the entire space. It guarantees the existence of a continuous linear functional that extends a given linear functional defined on a subspace.

Why is the Hahn Banach theorem important?

The Hahn Banach theorem is important because it allows us to extend functionals defined on subspaces to the entire space. This is useful in many areas of mathematics such as optimization, differential equations, and physics.

What is the intuition behind the Hahn Banach theorem?

The Hahn Banach theorem can be thought of as a way to "fill in the gaps" between a functional defined on a subspace and the entire space. It ensures that the extended functional is consistent with the original one on the subspace and is continuous on the entire space.

Can you provide an example of the Hahn Banach theorem in action?

One example of the Hahn Banach theorem in action is in the study of normed vector spaces. It guarantees the existence of a continuous linear functional that extends a given functional defined on a subspace to the entire space. This is useful in proving theorems about the dual space of a normed vector space.

What are some applications of the Hahn Banach theorem?

The Hahn Banach theorem has many applications in various areas of mathematics, including functional analysis, optimization, differential equations, and physics. It is also used in the study of Banach spaces and their dual spaces, as well as in the theory of convex sets and their extreme points.

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