- #1
muzak
- 44
- 0
Hi guys, two problems, first one I understand for the most part, the second one, I do not know how to set up and solve.
Let X = R[itex]^{n}[/itex] for x = (a[itex]_{1}[/itex],...,a[itex]_{n}[/itex]) and y = (b[itex]_{1}[/itex],...,b[itex]_{n}[/itex]), define
d[itex]_{\infty}[/itex](x,y) = max {|a[itex]_{1}[/itex]-b[itex]_{1}[/itex]|,...,|a[itex]_{n}[/itex]-b[itex]_{n}[/itex]|}. Prove that this is a metric.
Just the triangle inequality part for this one.
I've proven the first 3 properties, not quite sure on the last part.
My attempt was to break it up into n cases by supposing a single
difference as the max in each case but with ellipsis but I wasn't
sure on the exactly how. Here's what I attempted:
The distance between the two points is the largest of the n cases.
Let z = (z[itex]_{1}[/itex],...,z[itex]_{n}[/itex]). Then we have n cases to check.
Case 1: d(x,y) = |a[itex]_{1}[/itex]-b[itex]_{1}[/itex]|
Notice that |a[itex]_{1}[/itex]-z[itex]_{1}|[/itex]|[itex]\leq[/itex]max{|a[itex]_{1}[/itex]-z[itex]_{1}[/itex]|,
(I stopped here because I wasn't sure how to proceed,
should I write out to n cases with the ellipsis or two cases?)
What I was going to show after this was that the triangle inequality
holds for a[itex]_{1}[/itex], a[itex]_{2}[/itex], ..., a[itex]_{n}[/itex].
Prove that the set S = {(x[itex]_{1}[/itex],y[itex]_{1}[/itex] : x[itex]_{1}[/itex] + y[itex]_{1}[/itex] > 0}
is an open subset of R[itex]^{2}[/itex] in the Euclidean metric.
Euclidean metric, Schwarz Inequality?, Open Ball.
I'm not sure how to proceed with this one at all. Picture-wise,
it'd be the region above the y=-x line, I'm guessing,
and I'm guessing I have to pick some arbitrary point in that
region and calculate a strict inequality to show that this is open.
But I do not know how to proceed at all, I'm looking for some hand-holding
at this point really because I want to understand it step by step.
Thanks for any help!
Homework Statement
Let X = R[itex]^{n}[/itex] for x = (a[itex]_{1}[/itex],...,a[itex]_{n}[/itex]) and y = (b[itex]_{1}[/itex],...,b[itex]_{n}[/itex]), define
d[itex]_{\infty}[/itex](x,y) = max {|a[itex]_{1}[/itex]-b[itex]_{1}[/itex]|,...,|a[itex]_{n}[/itex]-b[itex]_{n}[/itex]|}. Prove that this is a metric.
Homework Equations
Just the triangle inequality part for this one.
The Attempt at a Solution
I've proven the first 3 properties, not quite sure on the last part.
My attempt was to break it up into n cases by supposing a single
difference as the max in each case but with ellipsis but I wasn't
sure on the exactly how. Here's what I attempted:
The distance between the two points is the largest of the n cases.
Let z = (z[itex]_{1}[/itex],...,z[itex]_{n}[/itex]). Then we have n cases to check.
Case 1: d(x,y) = |a[itex]_{1}[/itex]-b[itex]_{1}[/itex]|
Notice that |a[itex]_{1}[/itex]-z[itex]_{1}|[/itex]|[itex]\leq[/itex]max{|a[itex]_{1}[/itex]-z[itex]_{1}[/itex]|,
(I stopped here because I wasn't sure how to proceed,
should I write out to n cases with the ellipsis or two cases?)
What I was going to show after this was that the triangle inequality
holds for a[itex]_{1}[/itex], a[itex]_{2}[/itex], ..., a[itex]_{n}[/itex].
Homework Statement
Prove that the set S = {(x[itex]_{1}[/itex],y[itex]_{1}[/itex] : x[itex]_{1}[/itex] + y[itex]_{1}[/itex] > 0}
is an open subset of R[itex]^{2}[/itex] in the Euclidean metric.
Homework Equations
Euclidean metric, Schwarz Inequality?, Open Ball.
The Attempt at a Solution
I'm not sure how to proceed with this one at all. Picture-wise,
it'd be the region above the y=-x line, I'm guessing,
and I'm guessing I have to pick some arbitrary point in that
region and calculate a strict inequality to show that this is open.
But I do not know how to proceed at all, I'm looking for some hand-holding
at this point really because I want to understand it step by step.
Thanks for any help!