Graphing a function in a Normed linear space

[bugatti79]

Folks,

Could anyone give me a simple example on how to graph a function S each of the following normed linear spaces

i) R^2 with the taxi cab norm

ii) R^2 with the sup norm

iii) (R^2,|| ||) where x=(x_1,x_2) for some ||x|| is some linear function of x_1 and x_2

In other words, to give a a sample function S in \mathbb{N} so I can try to plot the above 3 request?

Thanks

 

[bugatti79]

I found something relating to unit balls on wiki at this link...http://en.wikipedia.org/wiki/Unit_sphere#Unit_balls_in_normed_vector_spaces

under the heading "unit balls in normed vector spaces"...how would one plot by hand the various p values ...ie p=1, p=1.414, p=2...etc?

thanks

 

[Greg Bernhardt]

Sure, I can give you a simple example for each of the three normed linear spaces you mentioned.

i) For R^2 with the taxi cab norm, also known as the Manhattan norm, the distance between two points (x1, y1) and (x2, y2) is defined as |x1-x2| + |y1-y2|. So, a sample function S in this space could be S(x,y) = |x| + |y|. To graph this, you can plot points on a grid and connect them with straight lines, as the taxi cab norm is based on the idea of traveling along city blocks.

ii) For R^2 with the sup norm, also known as the maximum norm, the distance between two points (x1, y1) and (x2, y2) is defined as max(|x1-x2|, |y1-y2|). A sample function S in this space could be S(x,y) = max(|x|, |y|). To graph this, you can plot points on a grid and connect them with straight lines, but the lines will be more diagonal than in the taxi cab norm.

iii) For (R^2, || ||) where x=(x1,x2) and ||x|| is some linear function of x1 and x2, a sample function S could be S(x,y) = x1 + 2x2. This is a simple linear function that satisfies the conditions. To graph this, you can plot points on a grid and connect them with straight lines, but the lines will be more diagonal than in the taxi cab norm.

I hope this helps! Let me know if you have any further questions.