Help me understand this proof of R^infinity's infinite-dimensionality

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In summary, the conversation discusses a proof that shows ℝ^∞ is infinite-dimensional. The proof involves assuming the existence of a finite base, but then contradicts this assumption by showing a vector that cannot be represented by the base vectors. This contradicts the initial assumption and proves that ℝ^∞ is indeed infinite-dimensional.
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JamesGold
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The proof goes as follows:

For contradiction, assume there exists |s|< ∞ such that s = {e1, e2, ... , en} and span(s} = ℝ^∞.

The above makes at least some sense to me. The proof goes on...

Let m > n and u = en+1 + en+2 + ... + em

u [itex]\notin[/itex] span(s), u [itex]\in[/itex] s

Because {e1, e2, ... , en} [itex]\notin[/itex] s, this implies a contradiction. Therefore ℝ^∞ is infinite-dimensional.

I don't see what we just did, and I don't see how what we just did proves that R^infinity is infinite-dimensional. Please help!
 
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  • #2
In words:
Assume that there is a finite base with a finite number of base vectors e1 to en*.
Consider en+1 (+maybe more vectors) - where it is unclear in which basis that is supposed to be*. This vector cannot be represented by a sum of e1 to en, therefore the initial assumption is wrong and a finite base cannot exist.

* I think the combination of those are a serious problem. While is possible to name (arbitrary) base vectors with ei, this cannot be extended to other vectors. If ei refer to the standard base, you cannot simply assume that a finite base uses those vectors.
 

1. How can Rinfinity have an infinite dimensionality?

Rinfinity is a mathematical concept that represents the set of all real numbers, including positive and negative numbers, fractions, and irrational numbers such as pi. Since there is an infinite number of possible values within this set, Rinfinity is considered to have infinite dimensionality.

2. What is the proof of Rinfinity's infinite dimensionality?

The proof of Rinfinity's infinite dimensionality can be demonstrated through Cantor's diagonal argument. This proof shows that, for any given set of real numbers, there will always be an uncountable number of numbers that are not contained within that set. Since Rinfinity contains all possible real numbers, it is uncountable and therefore has infinite dimensionality.

3. How does Rinfinity compare to other infinite-dimensional spaces?

Rinfinity is often used as a benchmark for infinite-dimensional spaces, as it represents the largest possible set of real numbers. Other infinite-dimensional spaces, such as Hilbert spaces or Banach spaces, have specific properties or restrictions that differentiate them from Rinfinity.

4. Can Rinfinity be physically represented?

No, Rinfinity cannot be physically represented or visualized, as it contains an infinite number of dimensions. It is a mathematical concept used to describe the set of all real numbers and does not have a physical manifestation.

5. Are there any real-world applications of Rinfinity?

While Rinfinity itself may not have direct real-world applications, the concept of infinite dimensionality is used in many fields of science and engineering. For example, in physics, infinite-dimensional spaces are used to describe the behavior of quantum systems, and in computer science, infinite-dimensional vectors are used in machine learning algorithms.

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