Proving Closure of Set of Operators w/ Property P Under Addition

[Crot]

Could you please give me a hint on how to show that a set of operators with a property P is closed under addition? In other words, how one could prove that a sum of any two operators from the set still possesses this property P. The set is assumed to be infinite.

Any references, comments, suggestions, etc. will be kindly appreciated.

 

[mfb]

It depends on the set of operators and on P.

One general concept: Take an arbitrary sum of two operators, show that the sum is an operator which satisfies P.

 

[Crot]

Thank you for your answer. But, this concept is really too general.
Are there any other techniques?

 

[mfb]

Your question is extremely general. Just one step more general would be "how do you prove a mathematical statement".

All techniques are some sort of this proof, as it is the definition of "closed".

 

[CellCoree]

To prove that a set of operators with a property P is closed under addition, you will need to show that the sum of any two operators from the set also possesses the property P. This can be done by using the definition of closure and the properties of P.

First, let's define what it means for a set of operators to be closed under addition. A set of operators is closed under addition if, for any two operators A and B in the set, their sum A+B is also in the set.

Now, to prove that the sum of any two operators from the set possesses the property P, we can use the following steps:

1. Start by assuming that A and B are two arbitrary operators from the set with property P.

2. Use the definition of P to show that A+B also satisfies property P. This may involve using the properties of P and the specific properties of A and B.

3. Since A and B were arbitrary operators from the set, this proof holds for any two operators from the set. Therefore, the sum of any two operators from the set must also possess property P.

4. This proves that the set of operators with property P is closed under addition.

In the case of an infinite set, you will need to use a proof by induction or use the fact that the infinite set can be represented as a limit of a sequence of finite sets.

I recommend consulting a textbook on functional analysis or linear algebra for more detailed explanations and examples of proving closure under addition for sets of operators. Additionally, you can also refer to research papers or online resources for specific examples and applications of this concept.