Compact Sets, Unit Balls, Norms, Inner Products: Delightful Reads

In summary, the conversation discussed the properties of the unit ball of a norm in a finite-dimensional vector space. The group considered different conditions for the unit ball, including convexity and symmetry, and discussed the relationship between norms and inner products. They also raised questions about the unit ball in infinite-dimensional vector spaces and those over arbitrary fields.
  • #1
qspeechc
844
15
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.
 
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  • #2


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.
 
  • #3


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.
 
  • #4


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.
 

1. What is a compact set?

A compact set is a subset of a metric space that is closed and bounded. This means that the set contains all of its limit points and can be contained within a finite distance. In other words, a compact set is a set that is both complete and finite.

2. What is a unit ball?

A unit ball is a subset of a metric space that is centered at the origin and has a radius of 1. In other words, it is a set of all points that are a certain distance (usually measured by a norm) from the origin. Unit balls are often used in the study of normed vector spaces.

3. What is a norm?

A norm is a mathematical function that assigns a positive length or size to a vector in a vector space. It measures the size or magnitude of a vector and is denoted by ||x||. Common examples of norms include the Euclidean norm and the maximum norm.

4. What is an inner product?

An inner product is a mathematical operation that takes two vectors and returns a scalar value. It is denoted by and has properties such as linearity and symmetry. Inner products are important in the study of vector spaces, as they allow for the definition of concepts such as orthogonality and angle between vectors.

5. How are compact sets, unit balls, norms, and inner products related?

Compact sets, unit balls, norms, and inner products are all important concepts in the study of functional analysis. Compact sets and unit balls are subsets of metric spaces, while norms and inner products are mathematical operations defined on vector spaces. They are related in the sense that norms and inner products can be used to define and study compact sets and unit balls, and can also be used to measure the size and distance of vectors within these sets.

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