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Compact Sets, Unit Balls, Norms, Inner Products and Other Topics to Delight Readers.

  1. Mar 12, 2010 #1
    Hi everyone. I wasn't really sure where to put this thread so I stuck it here, which seems the closest fit.

    Anyway I've been thinking about this for too long: what characterises the unit ball of a norm? Let's be specific: consider a finite-dimensional vector space, which may as well be [itex]\mathbb{R}^n[/itex].

    Now I think that every convex and compact set C containing the origin in its interior is the unit ball of some norm. I suspect convexity is too strong a condition and we can get away with less, but I can't think what weaker conditions will work. Define all the points lying on the boundary of the set C to have norm one. Now for any vector x in the space there must be some real number [itex]\lambda[/itex] such that [itex]\lambda x[/itex] lies on the boundary of C. Then define the norm of x to be [itex]|\lambda |^{-1}[/itex], and the norm of the origin to be zero. I think I have a proof that such a [itex]\lambda[/itex] does indeed exist; uniqueness may be troublesome but should follow from the convexity (this is why I want convexity); at least it is intuitively clear that this number exists and is unique. Now let's check this is actually a norm we have defined:

    1) It's clear the norm is always non-negative and zero only for the zero vector.

    2) The scaling of vectors (positive homogeneity) is all dandy with respect to the norm.

    3) The triangle inequality is tough. I have no idea why this should be true. Perhaps this follows from convexity too?

    Am I vaguely on the right track? What exactly characterises the unit balls of norm in a Euclidean space? Now every inner product induces a norm, so what characterises unit balls arising from inner-products? How do they differ from a unit ball of a norm that doesn't come from an inner-product? Are all norms from some inner-product?

    Already this matter brings up many questions, and there are even more if we generalise. What about if we think of a vector space of arbitrary dimension, not necessarily finite? I suspect we can say very little in this case, simply because almost none of the nice properties finite dimensional vector spaces have carries over to the infinite dimensional case, but I have nothing precise to say in this case. What about vector spaces over an arbitrary field? I suppose there are many other closely related questions, all very interesting, which I have not asked but I would like to know about as well.

    Any help? I know I've asked a lot of questions, so can you point me toward the literature so I can read up about this? Thank-you.
     
    Last edited: Mar 12, 2010
  2. jcsd
  3. Mar 12, 2010 #2

    LCKurtz

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    Re: Compact Sets, Unit Balls, Norms, Inner Products and Other Topics to Delight Reade

    For any norm, you must have ||-x|| = ||x||, which implies that the unit ball must be symmetric through the origin. So, for example, even in R2 your convex set and hence your unit ball can't be an equilateral triangle. Or any other triangle.
     
  4. Mar 12, 2010 #3
    Re: Compact Sets, Unit Balls, Norms, Inner Products and Other Topics to Delight Reade

    Yes, I recognised that minutes before I found your post. So then, is every compact set which is symmetric about the origin the unit ball of some norm? I dropped the convexity condition because it is too strong. The unit ball for the lp space is not convex for p<1:
    http://en.wikipedia.org/wiki/Unit_sphere
    I still haven't gotten any closer to answering these questions.
     
  5. Mar 12, 2010 #4

    LCKurtz

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    Re: Compact Sets, Unit Balls, Norms, Inner Products and Other Topics to Delight Reade

    But convexity is necessary. If x and y are on the unit ball,

    [tex]||tx + (1-t)y||\le ||tx||+||(1-t)y||\le t+1-t = 1\hbox{ for }0\le t\le 1[/tex]

    If p < 1 the lp spaces aren't normed.
     
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