# Boundedness in R^k

• Apr 20th 2010, 08:24 AM
rebghb
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 $\displaystyle A$ is bounded in $\displaystyle R$ if there exists two real numbers $\displaystyle a$ and $\displaystyle b$ such that $\displaystyle A\subset ]a;b[$.

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

Thanks!!
• Apr 20th 2010, 08:30 AM
Drexel28
Quote:

Originally Posted by rebghb
Hello everyone!

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

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

Thanks!!

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