- #1

AKG

Science Advisor

Homework Helper

- 2,565

- 4

[tex]d(x, y) = \left [\sum _{i = 1} ^n |x_i - y_i|^2\right ]^{1/2}[/tex]

Define metrics d

_{p}for each p in {1, 2, 3, ...} as follows:

[tex]d_p(x,y) = \left [\sum _{i = 1} ^n |x_i - y_i|^p\right ]^{1/p}[/tex]

Prove that each d

_{p}induces the same topology as the Euclidean metric.

To do this, I want to show that for every [itex]\epsilon > 0[/itex] and for every [itex]x \in \mathbb{R}^n[/itex], there is are [itex]\delta _1,\, \delta _2 > 0[/itex] such that for every [itex]y \in \mathbb{R}^n[/itex]:

[tex]\left [\sum _{i = 1} ^n |x_i - y_i|^2\right ]^{1/2} < \delta _1 \Rightarrow \left [\sum _{i = 1} ^n |x_i - y_i|^p\right ]^{1/p} < \epsilon[/tex]

and

[tex]\left [\sum _{i = 1} ^n |x_i - y_i|^p\right ]^{1/p} < \delta _2 \Rightarrow \left [\sum _{i = 1} ^n |x_i - y_i|^2\right ]^{1/2} < \epsilon[/tex]

Is this the right way to prove it? Where do I go from here? Induction on n, or p? Or maybe both? Or is there a way to do it without induction? Help would be very much appreciated!