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Thread: Unit circles of different norms- Subsets

  1. #1
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    Unit circles of different norms- Subsets

    Show that $\displaystyle B_1(0;1) \subset B_2(0;1) \subset B_{\infty}(0,1) $ in $\displaystyle \mathbb{R}^n$ for all $\displaystyle n$. I should clarify some notation here $\displaystyle B(0,1)$ means the open ball with center 0, and radius 1.

    Definition of norms:
    $\displaystyle d_1(x,y) := ||x - y||_1 = \displaystyle\sum_{j=1}^n|x_j - y_j|;$
    $\displaystyle d_2(x,y) := ||x - y||_2 = \left(\displaystyle\sum_{j=1}^n|x_j - y_j|^2\right)^{1/2}$;
    $\displaystyle d_{\infty}(x,y) := \displaystyle\max_{1\leq j \leq n}|x_j - y_j|$

    My attempt:
    I know that to show they are subsets of each other, I have to show the implication $\displaystyle \forall b \in B_1(0,1) \Rightarrow b \in B_2(0,1) $ as well and next $\displaystyle \forall b \in B_2(0,1) \Rightarrow b \in B_{\infty}(0,1)$ as well.

    So I have, let $\displaystyle b \in B_1(0,1)$, then $\displaystyle |b_1| + |b_2| + \cdots + |b_n| < 1$, I just don't see how this would imply $\displaystyle \left(|b_1|^2 + |b_2|^2 + \cdots + |b_n|^2\right)^{1/2} < 1$ or $\displaystyle |b_1|^2 + |b_2|^2 + \cdots + |b_n|^2 < 1$

    And I'm also unsure as to how to show $\displaystyle \forall b \in B_2(0,1) \Rightarrow b \in B_{\infty}(0,1)$.

    Hints?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by chrischen88 View Post
    Show that $\displaystyle B_1(0;1) \subset B_2(0;1) \subset B_{\infty}(0,1) $ in $\displaystyle \mathbb{R}^n$ for all $\displaystyle n$. I should clarify some notation here $\displaystyle B(0,1)$ means the open ball with center 0, and radius 1.

    Definition of norms:
    $\displaystyle d_1(x,y) := ||x - y||_1 = \displaystyle\sum_{j=1}^n|x_j - y_j|;$
    $\displaystyle d_2(x,y) := ||x - y||_2 = \left(\displaystyle\sum_{j=1}^n|x_j - y_j|^2\right)^{1/2}$;
    $\displaystyle d_{\infty}(x,y) := \displaystyle\max_{1\leq j \leq n}|x_j - y_j|$

    My attempt:
    I know that to show they are subsets of each other, I have to show the implication $\displaystyle \forall b \in B_1(0,1) \Rightarrow b \in B_2(0,1) $ as well and next $\displaystyle \forall b \in B_2(0,1) \Rightarrow b \in B_{\infty}(0,1)$ as well.

    So I have, let $\displaystyle b \in B_1(0,1)$, then $\displaystyle |b_1| + |b_2| + \cdots + |b_n| < 1$, I just don't see how this would imply $\displaystyle \left(|b_1|^2 + |b_2|^2 + \cdots + |b_n|^2\right)^{1/2} < 1$ or $\displaystyle |b_1|^2 + |b_2|^2 + \cdots + |b_n|^2 < 1$

    And I'm also unsure as to how to show $\displaystyle \forall b \in B_2(0,1) \Rightarrow b \in B_{\infty}(0,1)$.

    Hints?
    As $\displaystyle |b_i|<1, \ i=1,\ ..\ ,n$ we have $\displaystyle |b_i|^2 \le |b_i| $

    Hence:

    $\displaystyle \sum_i |b_i|^2\le \sum_i |b_i| <1$

    The same idea applies to the next part, if the sum of a number of positive terms is less than 1 each term individually is less than 1.

    CB
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