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Math Help - Boundedness in R^k

  1. #1
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    Boundedness in R^k

    Hello everyone!

    I'm new to the amazing world of analysis. I've got a question related to boundedness.
    A set A is bounded in R if there exists two real numbers a and b such that A\subset  ]a;b[.

    Well since we're talking about R^1, we can't differentiate between and open ball and a 1-cell, but when we're talking about boundedness in the complex plane R^2 for instance, must the set A of points P(x,y) be a_1 < x < a_2 and b_1 < y < b_2 or is it that d(p,q) = |p-q| < r for some  q??

    Thanks!!
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by rebghb View Post
    Hello everyone!

    I'm new to the amazing world of analysis. I've got a question related to boundedness.
    A set A is bounded in R if there exists two real numbers a and b such that A\subset [a,b][.

    Well since we're talking about R^1, we can't differentiate between and open ball and a 1-cell, but when we're talking about boundedness in the complex plane R^2 for instance, must the set A of points P(x,y) be a_1 < x < a_2 and b_1 < y < b_2 or is it that d(p,q) = |p-q| < r for some  q??

    Thanks!!
    There are multiple definitions. I mean, all you need is that A is contained in some bounded set. The real definition of boundedness is A\subseteq M\text{ is bounded}\Leftrightarrow \text{diam }A<\infty for \mathbb{R}^m (and other normed vector spaces) it is easier to think that A\text{ is bounded }\Leftrightarrow \|a\|<K,\text{ }\forall a\in A for some fixed K<\infty. So, A is bounded if it's either contained in some open ball B_{\delta}(0) or it's contained in some n-cell [a,b]^n
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