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Math Help - Unit circles of different norms- Subsets

  1. #1
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    Unit circles of different norms- Subsets

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

    Definition of norms:
    d_1(x,y) := ||x - y||_1 = \displaystyle\sum_{j=1}^n|x_j - y_j|;
    d_2(x,y) := ||x - y||_2 = \left(\displaystyle\sum_{j=1}^n|x_j  - y_j|^2\right)^{1/2};
    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 \forall b \in B_1(0,1) \Rightarrow b \in B_2(0,1) as well and next \forall b \in B_2(0,1) \Rightarrow b \in B_{\infty}(0,1) as well.

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

    And I'm also unsure as to how to show \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 B_1(0;1) \subset B_2(0;1) \subset B_{\infty}(0,1) in \mathbb{R}^n for all n. I should clarify some notation here B(0,1) means the open ball with center 0, and radius 1.

    Definition of norms:
    d_1(x,y) := ||x - y||_1 = \displaystyle\sum_{j=1}^n|x_j - y_j|;
    d_2(x,y) := ||x - y||_2 = \left(\displaystyle\sum_{j=1}^n|x_j - y_j|^2\right)^{1/2};
    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 \forall b \in B_1(0,1) \Rightarrow b \in B_2(0,1) as well and next \forall b \in B_2(0,1) \Rightarrow b \in B_{\infty}(0,1) as well.

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

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

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

    Hence:

    \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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