Dtjytyk

How to test the equality of two Pearson correlation coefficients computed from the same sample?

How can I protect witches in combat who wear limited clothing?

Didn't get enough time to take a Coding Test - what to do now?

Difference between "generating set" and free product?

What's the point in a preamp?

Can a novice safely splice in wire to lengthen 5V charging cable?

Can a 1st-level character have an ability score above 18?

Is every episode of "Where are my Pants?" identical?

How do I add random spotting to the same face in cycles?

Is this wall load bearing? Blueprints and photos attached

Typeface like Times New Roman but with "tied" percent sign

Segmentation fault output is suppressed when piping stdin into a function. Why?

Did God make two great lights or did He make the great light two?

Do warforged have souls?

How is simplicity better than precision and clarity in prose?

Can undead you have reanimated wait inside a portable hole?

Are spiders unable to hurt humans, especially very small spiders?

How to delete random line from file using Unix command?

Road tyres vs "Street" tyres for charity ride on MTB Tandem

system() function string length limit

Can smartphones with the same camera sensor have different image quality?

How does ice melt when immersed in water?

Was credit for the black hole image misattributed?

Can the DM override racial traits?

In what sense is the L2 norm the canonical norm on $mathbb{R}^n$ but the max-norm is the canonical norm on $mathbb{Q}_p^n$?

1

1

$begingroup$

I understand that the L2 norm is natural to consider on $mathbb{R}^n$ both because of its geometric intuition, and because it is induced by the dot product on $mathbb{R}^n$. I also understand that a max-norm is convenient to use on $mathbb{Q}_p^n$ because it extends the nonarchimedean triangle inequality to the vector space. I further understand that every norm on a vector space over a normed field is topologically equivalent.

However, geometrically, different norms can have very different structures. Is there a natural criterion of "niceness" which, when applied to real vector fields, singles out an L2-norm (with respect to some basis), and when applied to $p$-adic vector spaces, singles out the max-norm (with respect to some basis)?

If it helps, this question was motivated by some comments in Shai Haran's "Mysteries of the Real Prime", where he noted that (following Tate) the factors of the completed Riemann zeta function can be viewed as integrals over the places of $mathbb{Q}$, and in fact the integrand may be viewed as a limit of a function induced by a norm on $mathbb{Q}_p^2$ or $mathbb{R}^2$. But norm Haran used on $mathbb{Q}_p^2$ was the max-norm, and the norm he used on $mathbb{R}^2$ was the L2 norm. I am searching for a principled justification for this choice beyond "Well, it gave us the result we wanted."

real-analysis vector-spaces p-adic-number-theory

share|cite|improve this question

asked Mar 22 at 20:20

Joseph GreenwoodJoseph Greenwood

707

$endgroup$

add a comment | 

1

1

$begingroup$

I understand that the L2 norm is natural to consider on $mathbb{R}^n$ both because of its geometric intuition, and because it is induced by the dot product on $mathbb{R}^n$. I also understand that a max-norm is convenient to use on $mathbb{Q}_p^n$ because it extends the nonarchimedean triangle inequality to the vector space. I further understand that every norm on a vector space over a normed field is topologically equivalent.

However, geometrically, different norms can have very different structures. Is there a natural criterion of "niceness" which, when applied to real vector fields, singles out an L2-norm (with respect to some basis), and when applied to $p$-adic vector spaces, singles out the max-norm (with respect to some basis)?

If it helps, this question was motivated by some comments in Shai Haran's "Mysteries of the Real Prime", where he noted that (following Tate) the factors of the completed Riemann zeta function can be viewed as integrals over the places of $mathbb{Q}$, and in fact the integrand may be viewed as a limit of a function induced by a norm on $mathbb{Q}_p^2$ or $mathbb{R}^2$. But norm Haran used on $mathbb{Q}_p^2$ was the max-norm, and the norm he used on $mathbb{R}^2$ was the L2 norm. I am searching for a principled justification for this choice beyond "Well, it gave us the result we wanted."

real-analysis vector-spaces p-adic-number-theory

share|cite|improve this question

asked Mar 22 at 20:20

Joseph GreenwoodJoseph Greenwood

707

$endgroup$

add a comment | 

$begingroup$

I understand that the L2 norm is natural to consider on $mathbb{R}^n$ both because of its geometric intuition, and because it is induced by the dot product on $mathbb{R}^n$. I also understand that a max-norm is convenient to use on $mathbb{Q}_p^n$ because it extends the nonarchimedean triangle inequality to the vector space. I further understand that every norm on a vector space over a normed field is topologically equivalent.

However, geometrically, different norms can have very different structures. Is there a natural criterion of "niceness" which, when applied to real vector fields, singles out an L2-norm (with respect to some basis), and when applied to $p$-adic vector spaces, singles out the max-norm (with respect to some basis)?

If it helps, this question was motivated by some comments in Shai Haran's "Mysteries of the Real Prime", where he noted that (following Tate) the factors of the completed Riemann zeta function can be viewed as integrals over the places of $mathbb{Q}$, and in fact the integrand may be viewed as a limit of a function induced by a norm on $mathbb{Q}_p^2$ or $mathbb{R}^2$. But norm Haran used on $mathbb{Q}_p^2$ was the max-norm, and the norm he used on $mathbb{R}^2$ was the L2 norm. I am searching for a principled justification for this choice beyond "Well, it gave us the result we wanted."

real-analysis vector-spaces p-adic-number-theory

share|cite|improve this question

asked Mar 22 at 20:20

Joseph GreenwoodJoseph Greenwood

707

$endgroup$

share|cite|improve this question

asked Mar 22 at 20:20

Joseph GreenwoodJoseph Greenwood

707

share|cite|improve this question

asked Mar 22 at 20:20

Joseph GreenwoodJoseph Greenwood

707

asked Mar 22 at 20:20

Joseph GreenwoodJoseph Greenwood

707

asked Mar 22 at 20:20

Joseph GreenwoodJoseph Greenwood

707