How to interpret absolute value bars?

[aj-smith]

I was further looking into the Cauchy schwarz inequality and i got to a statement as follows:

A·B ≤ |A·B|

However, when I tried to prove this using numbers on paper, I wasn't sure if the absolute value bars distribute among each term, which would lead to |A|·|B|, or if the final product is then absolute.

I was wondering how you would interpret the above statement (assuming all capital letters represent vectors

These are the components I assumed.

A= {1,2} B= {-2,3} c= -2

Also how would the following below be interpreted?

|cA·B|

Thank you.

 

[DonAntonio]

I was further looking into the Cauchy schwarz inequality and i got to a statement as follows:

A·B ≤ |A·B|

*** I presume you're talking of real inner product spaces since otherwise (i.e., in a complex space) this has no meaning. ***


However, when I tried to prove this using numbers on paper, I wasn't sure if the absolute value bars distribute among each term, which would lead to |A|·|B|, or if the final product is then absolute.


*** You seemed to be using the dot $\cdot$ to represent inner product of two vectors, so $|A|\cdot |B|$ would be the

product of two real NUMBERS, IF by $|A|$ you mean the norm of vector A...something completely different. ***



I was wondering how you would interpret the above statement (assuming all capital letters represent vectors

These are the components I assumed.

A= {1,2} B= {-2,3} c= -2

Also how would the following below be interpreted?

|cA·B|


*** This is the abs. value of the inner product of the vector cA by the vector B, i.e. $$|(-2,-4)\cdot (-2,3)|=|4-12|=8$$

DonAntonio


Thank you.

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