Complex Inner-Product Spaces: True or False?

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Homework Help Overview

The discussion revolves around properties of complex inner-product spaces, specifically evaluating which statements about inner products are universally true or false. Participants are examining the validity of several properties related to inner products in this mathematical context.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are analyzing specific statements about inner products, questioning the implications of complex constants, and discussing the conditions under which certain inequalities hold. There is a focus on the potential typo in one of the statements and its relation to known inequalities.

Discussion Status

Some participants have offered insights into the nature of the statements, particularly regarding the conjugation of complex constants and the validity of the Cauchy-Schwarz inequality. There is an ongoing exploration of the implications of these properties without reaching a definitive consensus.

Contextual Notes

Participants note that the interpretation of conjugation may vary depending on the context of the text being referenced, indicating a need for clarification based on the source material.

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Homework Statement



In an arbitrary complex inner-product space V which of the following is not always true?

a. <αu + βv, w> = α<u, w> + β<v, w>
b. |<u, v>|2 ≤ <u, u> <v, w>
c. <u, αv> = α<u, v>
d. <0, u> = 0

Homework Equations


None


The Attempt at a Solution



The correct answer must be a, but I don't know what to do with b.

a. False. If the constants are complex, they will be conjugated.
b. Any ideas?
c. True. Constants in the second argument will not be conjugated.
d. True. Multiplying a complex number by zero still results in a zero.
 
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If b) were |<u,v>|^2<=<u,u>*<v,v> it would be the Cauchy-Schwarz inequality. It's true. Is the 'w' a typo?
 
Dick said:
If b) were |<u,v>|^2<=<u,u>*<v,v> it would be the Cauchy-Schwarz inequality. It's true. Is the 'w' a typo?

Well, my professor has a 'w' on the assignment, but you must be right.

Thanks again!
 
Which argument gets the conjugate depends on whether it is a physics text or a math text. So consult the text.
 

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