Results 1 to 8 of 8

Thread: Euclidean Spaces

  1. #1
    Member
    Joined
    Jun 2008
    Posts
    170

    Euclidean Spaces

    Let's say if we have $\displaystyle \bold{x} \in \mathbb{R}^{n} $ and we want to show that $\displaystyle B_{r}(\bold{x}) $(open ball of radius $\displaystyle r >0 $) is convex. We know that $\displaystyle d(\bold{x}, \bold{y}) = |\bold{x}- \bold{y}| $ (e.g. the norm).

    We can still apply the triangle inequality right? So we have: $\displaystyle |t(\bold{y}-\bold{x})+(1-t)(\bold{z}-\bold{x})| \leq t|\bold{y}-\bold{x}| + (1-t)|\bold{z}-\bold{x}| < r $.


    In other words, does $\displaystyle d(\bold{x}, \bold{y}) = |\bold{x}- \bold{y}| $ only apply to $\displaystyle \mathbb{R} $? Or does it apply to general $\displaystyle \mathbb{R}^{n} $ so that we can invoke the triangle inequality? We can use the triangle inequality to prove the general case (which is what I did) right?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,742
    Thanks
    2814
    Awards
    1
    Quote Originally Posted by particlejohn View Post
    In other words, does $\displaystyle d(\bold{x}, \bold{y}) = |\bold{x}- \bold{y}| $ only apply to $\displaystyle \mathbb{R} $? Or does it apply to general $\displaystyle \mathbb{R}^{n} $ so that we can invoke the triangle inequality? We can use the triangle inequality to prove the general case (which is what I did) right?
    Perhaps I should not attempt an answer because I may have missed your point.
    But one of the requirements for a metric is the triangle inequality holds.
    That is, if $\displaystyle d$ is a metric on a space then $\displaystyle d(x,y) \leqslant d(x,z) + d(z,y)$ for all x, y, & z in the space.
    Does that get at what you are asking?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jun 2008
    Posts
    170
    yes, thanks.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Jun 2008
    Posts
    170
    how about the euclidean norm? does it hold for that?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,742
    Thanks
    2814
    Awards
    1
    Quote Originally Posted by particlejohn View Post
    how about the euclidean norm? does it hold for that?
    Yes it does. Here is how that goes.
    $\displaystyle \left\{ {y,z} \right\} \subseteq B_r (x)\; \Rightarrow \;\left\| {y - x} \right\| < r\;\& \;\left\| {z - x} \right\| < r$
    $\displaystyle \begin{array}{*{20}c}
    {\left\| {\lambda y\_(1 - \lambda )z - x} \right\|} & = & {\left\| {\lambda (y - x) + (1 - \lambda )(z - x)} \right\|} \\
    {} & \leqslant & {\lambda \left\| {y - x} \right\| + (1 - \lambda )\left\|{z - x} \right\|} \\ {} & < & {\lambda r + (1 - \lambda )r} \\ {} & = & r \\ \end{array} $
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member
    Joined
    Jun 2008
    Posts
    170
    so the triangle inequality applies to the following: $\displaystyle |\bold{x}| = \sqrt{x_{1}^{2} + \cdots x_{n}^{2}} $?

    The triangle inequality works for all metrics on Euclidean spaces? But not on say $\displaystyle L^{p} $ spaces?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,742
    Thanks
    2814
    Awards
    1
    I think that you are confusing yourself: the idea of a norm on a linear space with the distance function or metric.
    Have a look at these pages:
    Norm -- from Wolfram MathWorld
    Vector Norm -- from Wolfram MathWorld
    In the case of a ‘norm’ as with a metric, the triangle inequality is a requirement.
    I think that you need to go to the instructor and clear away this confusion.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Member
    Joined
    Jun 2008
    Posts
    170
    yeah even Rudin agrees.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Any metric spaces can be viewed as a subset of normed spaces
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: Dec 15th 2011, 03:00 PM
  2. non-euclidean
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Dec 3rd 2010, 04:57 AM
  3. Bases in Locally Euclidean spaces
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Mar 30th 2010, 07:40 PM
  4. Replies: 3
    Last Post: Jun 1st 2008, 01:51 PM
  5. Replies: 1
    Last Post: May 22nd 2008, 11:31 AM

Search Tags


/mathhelpforum @mathhelpforum