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June 18th, 2018, 11:10 AM  #1 
Newbie Joined: Dec 2012 Posts: 12 Thanks: 0  Norms and the Cauchy Schwartz Inequality
I'm having a bit of an issue with the conceptual idea of Norms. I understand that it is used to define distances in $\displaystyle R^n$. So my questions are:

June 18th, 2018, 02:24 PM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 443 Thanks: 254 Math Focus: Dynamical systems, analytic function theory, numerics 
1. The pnorm has nothing to do with the dimension. Different choices of $p$ define different ways of measuring distance. An analogy (which you should not read too much into beyond some intuition) is to consider two points which lie on a cylinder. One way to measure the distance between them is to simply connect them with a line and measure the Euclidean length of that line. However, notice that this path from one point to another does not remain on the surface of the cyclinder. Alternatively, you could measure the distance by computing the shortest path between these points which remains on the cyclinder. Neither is correct or wrong, they are just different. In this way, different norms for vectors give different answers which are neither wrong nor correct. Actually, it turns out that for finite dimensional vector spaces, all norms are equivalent. 2. Your friend is wrong. I'm not even sure what he/she is trying to say. If $X,Y$ are vectors of different dimension, then they live in different vector spaces. So what would $X+Y$ even mean? CauchySchwarz is also not a good idea since it applies to inner product spaces only which is less general than a normed vector space. I think you should google the triangle inequality. 
June 18th, 2018, 11:40 PM  #3 
Senior Member Joined: Oct 2009 Posts: 494 Thanks: 164 
For the 1norms, this is less relevant, but for pnorms with $1<p<+\infty$, there is the Holder inequality which is a generalization of CauchySchwartz.

June 20th, 2018, 02:42 PM  #4 
Newbie Joined: Dec 2012 Posts: 12 Thanks: 0 
Thanks a lot, that helps to clarify things a bit. I think my issue might be that I need to take a pure linear algebra course, as opposed to courses that apply linear algebra...It might help with these conceptual issues.


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cauchy, inequality, norms, schwartz 
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