Kreiszig 1.3-9: l-infinity is not seperable

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In summary, Kreiszig's example 1.3-9 shows that the metric space l^{\infty} is not separable by considering the bijection between the interval [0,1] and the set Y of sequences, y = (\eta_1, \eta_2, ...), whose terms consist of only 1's and 0's. By showing that Y is uncountable and that each ball in Y contains an element of a dense set M in l^{\infty}, it is concluded that M must also be uncountable. This means that l^{\infty} does not have a dense subset that is countable, thus proving that it is not separable.
  • #1
Rasalhague
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Kreiszig: Introductory Functional Analysis with Applications, example 1.3-9, states that the metric space [itex]l^{\infty}[/itex] is not seperable. He invites the us to consider the bijection between the interval [0,1] and the set Y of sequences, [itex]y = (\eta_1, \eta_2, ...)[/itex], whose terms consist of no number other than a 1 or a 0:

[tex]\hat{y}\in \left [ 0,1 \right ] = \sum_{k=1}^{\infty} \frac{\eta_k}{2^k}.[/tex]

The existence of this bijection, and the fact that [0,1] is uncountable, shows that Y is uncountable.

Consider the uncountably many balls of radius 1/3, each centered on one y in Y. (Whether open or closed is unstated, so I guess that's not relevant.) These balls don't intersect.

If M is any dense set in in [itex]l^{\infty}[/itex], each of these nonintersecting balls must contain an element of M.

Therefore M is uncountable. M was chosen arbitrarily. Therefore, he concludes, [itex]l^{\infty}[/itex] has no dense subset which is countable. In other words, [itex]l^{\infty}[/itex] is not seperable.

My question. Am I right in thinking this assumes that no ball can contain a limit point of M which is not in M? If so, why is that?
 
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  • #2
Rasalhague said:
My question. Am I right in thinking this assumes that no ball can contain a limit point of M which is not in M? If so, why is that?

I'm a bit confused on why you would think of limit points here. You suppose M is dense, right? Well, in that case, every point of of [tex]\ell^\infty[/tex] (which is not in M) is a limit point of M! So that's not what's going on here.

Can you tell us where you're stuck in the proof??
 
  • #3
Also note that the function you mention (i.e. the one that sends [tex]\hat{y}[/tex] to [tex]\sum{\eta_k/2^k}[/tex] ) is NOT a bijection. It is merely a surjection! It is indeed true that a point in [0,1] can have two binary representations:

[tex]\frac{1}{2}=\frac{1}{2}+0+0+0+0+\hdots=\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+...[/tex]
 
  • #4
Re. #3. Ah, yes, thanks for pointing that out. I thought, for some reason, he was saying that Y is uncountable because it has the same cardinality as [0,1], but I see now he actually only needs to show that it has at least as great a cardinality as [0,1]. (Incidentally, do you know if there exists a bijection between them?)

Re. #2. Got it now! (I hope.)

There are uncountably many balls, and each contains an element of l-infinity, namely each y in Y. In each ball, either its y is in M, or its y is a limit point of M. So if its y is not in M, then this y is a limit point of M, so every open ball around an element of Y, including the ball indicated, must contain an element of M. Therefore every ball of radius, say, 1/3 around some element of Y must contain an element of M. Therefore M too has at least as great a cardinality as [0,1], and so is uncountable.

I think what was confusing me was only that I'm new to some of the concepts and they hadn't bedded down yet in my brain. I just didn't follow through the reasoning far enough.
 
  • #5
Yes, that is both correct. :biggrin:

And it is indeed true that there is a bijection between [0,1] and the subset of [tex]\ell^\infty[/tex] that you mention. However, finding that explicit bijection is not easy...
 
  • #6
Personally, I despise decimal expansions, and I try to avoid them whenever I can.

In this case, just take [itex]Y=\{0,1\}^\mathbb{N}[/itex], the set of sequences consisting of 0's and 1's. This is an uncountable subset of \ell\infty, with any pair distinct elements a distance of 1 from each other. The uncountability is even easier now: there is an obvious bijection between Y and the power set of [itex]\mathbb{N}[/itex].

Now suppose Z is a dense subset. By definition this means that any ball around any element meets Z. In particular, for every y in Y the ball B(y) at y of radius 1/2 contains an element z(y) of Z. But if y' in Y is another element, then B(y') and B(y) are disjoint (as ||y-y'||=1), so z(y)/=z(y'). Hence the map Y->Z, y\mapsto z(y) is an injection . As Y is uncountable, Z must be too.

(I find this argument also a bit clearer than talking about 'uncountably many balls')
 
  • #7
Thanks for the confirmation micromass, and thanks Landau for the extra insight. That's very helpful.
 

What does "Kreiszig 1.3-9" refer to?

"Kreiszig 1.3-9" refers to a specific theorem in the field of functional analysis, which states that the l-infinity space is not separable.

What is l-infinity space?

l-infinity space is a mathematical concept that represents a space of infinite sequences of real numbers that are bounded above by a certain value. It is commonly used in functional analysis and other areas of mathematics.

What does it mean for a space to be separable?

A separable space is one that contains a countable dense subset, meaning that there exists a subset of elements that are "dense" or "closely packed" within the space. This is an important property in mathematics and has applications in fields such as topology and measure theory.

Why is it important that l-infinity is not separable?

The fact that l-infinity is not separable has important implications in functional analysis, as it means that certain properties and theorems that hold true for separable spaces may not hold true for l-infinity. This highlights the importance of understanding the properties and limitations of different types of mathematical spaces.

What are some potential real-world applications of this theorem?

This theorem has applications in fields such as signal processing and data compression, as well as in the study of certain types of mathematical spaces. It also has implications for understanding the structure and properties of infinite-dimensional spaces, which are relevant in many areas of mathematics and physics.

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