The Annihilator of a Set ....Remarks by Garling After Proposition 11.3.5 ....

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In summary, the Annihilator of a Set is a mathematical concept that refers to the set of all elements that "annihilate" the original set by multiplying with the identity element. To calculate it, one must first find the identity element for the set's operation, then find the elements that result in the identity element when multiplied. This concept is significant as it is a subgroup of the original set's operation and follows the same rules and properties. Examples of the Annihilator include sets of real numbers and matrices. It is used in fields such as abstract algebra, group theory, and vector spaces to understand mathematical structures and operations.
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I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ...

I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ...

I need some help in order to understand the meaning and the point or reason for some remarks by Garling made after Proposition 11.3.5 ...

The remarks by Garling made after Proposition 11.3.5 ... read as follows:View attachment 8966I understand the "mechanics" of the equations/expressions in Garling's remarks but do not know the reason or the point of his remarks ... can someone please explain the reasons behind or the point of Garling's remarks ... further what does he mean by "decomposition" ... ... Help will be appreciated ...

Peter==========================================================================================

The above post refers to Proposition 11.3.5 ... so I am providing text of the same in order for readers to be able to understand the context of my question ...View attachment 8967Hope that helps ...

Peter
 

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Peter said:
I understand the "mechanics" of the equations/expressions in Garling's remarks but do not know the reason or the point of his remarks ... can someone please explain the reasons behind or the point of Garling's remarks ... further what does he mean by "decomposition" ... ...

Garling is saying that any vector $z\in V$ can be written uniquely as $z=\lambda x+y$ where $\lambda$ is a scalar, $x$ is a unit vector and $y\in x^\bot$.
 
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Hi Peter,

I haven't read the book you mentioned, but I can try to help you understand the remarks made by Garling in Proposition 11.3.5. From what I understand, Garling is discussing the concept of decomposition in metric spaces and normed spaces.

In mathematics, decomposition refers to breaking down a complex object or structure into simpler components. In the context of metric and normed spaces, it means finding a way to express a given space as a combination of simpler spaces. This can be useful in understanding the properties and structure of the space.

In Proposition 11.3.5, Garling is showing that every normed space can be decomposed into a direct sum of two subspaces, one of which is finite-dimensional and the other is infinite-dimensional. This result is important because it tells us that every normed space has a certain "structure" that can be broken down into simpler components.

The reason for Garling's remarks after the proposition may be to provide some intuition or explanation for why this decomposition is useful and what it tells us about the space. Without further context, it's hard to say exactly what he means by "mechanics" and "point" in this context, but I hope my explanation of decomposition has helped you understand the general idea.

If you have any specific questions about the equations or expressions in Garling's remarks, I suggest looking for more context in the book or asking for clarification from your instructor or classmates. Understanding mathematical concepts can be challenging, but with patience and practice, you'll get there. Good luck with your studies!
 

FAQ: The Annihilator of a Set ....Remarks by Garling After Proposition 11.3.5 ....

1. What is the "Annihilator of a Set" in mathematics?

The annihilator of a set in mathematics is the set of all elements that, when multiplied with any element in the original set, result in the identity element of the operation being used. In other words, it is the set of all elements that "annihilate" or cancel out the elements in the original set.

2. How is the annihilator of a set useful in mathematics?

The annihilator of a set is useful in various areas of mathematics, such as abstract algebra and functional analysis. It helps in defining and understanding concepts like subspaces, ideals, and dual spaces.

3. Can the annihilator of a set be empty?

Yes, the annihilator of a set can be empty. This happens when the set being considered is the whole space or contains the identity element of the operation being used.

4. How is the annihilator of a set related to the null space of a matrix?

The annihilator of a set is closely related to the null space of a matrix. In fact, the null space of a matrix is the annihilator of the row space of that matrix.

5. What is the significance of Proposition 11.3.5 in relation to the annihilator of a set?

Proposition 11.3.5 states that the annihilator of a set is always a subspace of the dual space of the vector space containing the original set. This is significant because it helps in understanding and proving various properties and theorems related to the annihilator of a set.

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