Limited operators - continuity?

In summary, the conversation discusses understanding a theorem and its analysis terms, specifically the concepts of a limited operator and a strongly continuous application. The speaker expresses a desire for an intuitive interpretation and a simple understanding without strict studying. The conversation also touches on the difficulty of finding the appropriate place to discuss functional analysis and someone suggests checking out the Wikipedia entry on "bounded operator" for an easy proof.
  • #1
diegzumillo
173
18
Hi there! :)

I'm trying to understand a theorem, but it's full with analysis (or something) terms unfamiliar to me.

Is there an intuitive interpretation for the sentence: 'An operator being limited is equivalent to continuity in the topolgy of the norm'?

Also, how can I partially understand what is a "strongly continuous application"?

I understand that for a plain comprehension, one is required to follow a strict study on these subjects. But right now I'm happy with a simple intuition.

Edit: Is this the right place? I'm not sure analysis is the subject here!
 
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  • #2
Check out the Wikipedia entry on "bounded operator". It contains an easy proof.

I agree that it's hard to figure out where to put a post about functional analysis. Strangerep and I were just saying that in another thread.
 
  • #3
Thanks Fredrik!

(I thought I subscribed to thread!)
 

What are limited operators?

Limited operators are mathematical operators that have certain limitations or restrictions on their use. They are typically used in the context of calculus and are defined as operators that map a function to another function within a specific domain.

What is continuity?

Continuity is a mathematical concept that describes the smoothness and connectedness of a function. A function is considered continuous if there are no sudden jumps or breaks in its graph. This means that the function can be drawn without lifting the pen from the paper.

How can limited operators affect continuity?

Limited operators can affect continuity by imposing restrictions on the types of functions that can be used in their applications. For example, a limited operator may only work on continuous functions, which means that any function that is not continuous will not be valid for the operator.

What are some common examples of limited operators?

Some common examples of limited operators include the derivative, integral, and limit operators. These operators have specific rules and restrictions on their use, such as only being applicable to continuous functions or only being defined for certain values.

Why is understanding limited operators and continuity important?

Understanding limited operators and continuity is important for various reasons. It allows us to accurately model and analyze real-world phenomena, make predictions and solve problems in mathematics, and understand the behavior of functions in different contexts. It also provides the foundation for advanced mathematical concepts and applications in fields such as physics, engineering, and economics.

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