How is this inequality obtained?

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    Inequality Integral
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Discussion Overview

The discussion revolves around understanding an inequality presented in a text on operator theory, specifically how it is derived and its implications. Participants express confusion regarding the connection between the inequality, the Cauchy-Schwarz inequality, and the properties of integrals and inner products.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants question the derivation of the inequality in the text, suggesting it resembles the Cauchy-Schwarz inequality but expressing uncertainty about this connection.
  • There is a request for clarification on how the norm of the integral relates to the supremum of an inner product.
  • Participants inquire whether demonstrating that the integral is bounded is necessary and if this serves as motivation for the inequality.
  • Several participants note discrepancies in the Google Books preview, with some unable to access the relevant pages, leading to requests for more explicit context regarding the problem.
  • One participant mentions that the inequality involves complex notation that requires further explanation to be understood.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the derivation of the inequality or its implications, with multiple competing views and uncertainties expressed throughout the discussion.

Contextual Notes

Limitations include varying access to the Google Books preview, which affects participants' ability to engage with the material directly. There is also a lack of clarity regarding the specific notation and definitions used in the inequality.

Boromir
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Introduction to Operator Theory and Invariant Subspaces - B. Beauzamy - Google BooksIn page 144 of this preview I don't know how they obtain the inequality in (1). It looked like cauchy schwarz but I don't think it is.

I also don't know how they connect the norm of the integral to the supremum of an inner product.

Finally, is it necessary to show the integral is bounded? Is that the motivation?
 
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Re: integral inequality

Boromir said:
Introduction to Operator Theory and Invariant Subspaces - B. Beauzamy - Google BooksIn page 144 of this preview I don't know how they obtain the inequality in (1). It looked like cauchy schwarz but I don't think it is.

I also don't know how they connect the norm of the integral to the supremum of an inner product.

Finally, is it necessary to show the integral is bounded? Is that the motivation?
On my computer, the Google Books preview of this text only goes up to p.139. Please state the problem and its context more explicitly.
 
Re: integral inequality

Opalg said:
On my computer, the Google Books preview of this text only goes up to p.139. Please state the problem and its context more explicitly.

How can it be different for different computers?
 
Re: integral inequality

Boromir said:
How can it be different for different computers?

On my computer pages 144-151 are not shown in the preview. It would really be best if you took the time to state the problem yourself.
 
Re: integral inequality

Boromir said:
Opalg said:
On my computer, the Google Books preview of this text only goes up to p.139. Please state the problem and its context more explicitly.

How can it be different for different computers?
It depends entirely on what Google's server chooses to send to different users.
 
Here is what the inequality says. It seems like a bunch of notation is required to be explained to make sense of this. But it seems like Opalg said just CS inequality.

$$
\left| \left< (\int |f| ~ dE)x,y\right> \right|^2 \leq \left< (\int |f| ~ dE)x,x \right> \left< \int (|f| ~ dE)y,y\right> $$
 

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