Space of continuous functions C[a,b]

In summary, the conversation discusses the dimension of the vector space C[a,b] and whether its Hamel basis is countable or uncountable. Different proofs and arguments are presented, including using Baire's category theorem and counting arguments. It is concluded that the Hamel basis of C[a,b] is either finite or uncountable, with an example of an uncountable linearly independent subset provided. Additionally, it is mentioned that the space of all real sequences which are eventually zero has a countable Hamel basis.
  • #1
ninty
12
0
We know that dim(C[a,b]) is infinte. Indeed it cannot be finite since it contains the set of all polynomials.
Is the dimension of a Hamel basis for it countable or uncountable?

I guess if we put a norm on it to make a Banach space, we could use Baire's to imply uncountable.
I am however interested in a proof that does not rely on equipping the vector space with a norm.

If it's too long winded but available somewhere else(books, articles) kindly point it out to me and I'll read it.
 
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  • #2
Hamel bases are always finite or uncountable.
 
  • #3
JSuarez said:
Hamel bases are always finite or uncountable.

That's true if your space is an infinite dimensional normed vector space which is complete, and that result follows from Baire's category theorem AFAIK. So you haven't really strayed from the original proof.

A counting argument might suffice. Suppose there was a countable basis. There are only continuum many choices of finite linear combinations from these, but the cardinality of the vector space is larger than that
 
  • #4
I used instead the fact that exponentials are continuous
Then E:={e^ax : a real } is a subset of C[a,b]

It's clear that for finite n, {e^ix : i in N} is linearly independent.
I'm fudging, but that would seem to imply that E is also linearly independent, since every finite subset is linearly independent.
Hence cardinality of a basis is at least |E| = continuum
 
  • #5
Office_Shredder said:
A counting argument might suffice. Suppose there was a countable basis. There are only continuum many choices of finite linear combinations from these, but the cardinality of the vector space is larger than that

False: the cardinality of C[a,b] is the cardinality of the continuum. (Each function in C[a,b] is uniquely determined by its values on the rational numbers in [a,b].)

But yes, you could show that E = {eax | a real} is an uncountable linearly independent subset of C[a,b], to show that C[a,b] has no countable Hamel basis.

JSuarez said:
Hamel bases are always finite or uncountable.

The space of all real sequences which are eventually zero has a countable Hamel basis.
 
Last edited:

What is the "Space of continuous functions C[a,b]"?

The space of continuous functions C[a,b] is a mathematical concept used in the field of analysis to describe a set of functions that are continuous over a specific interval [a,b]. It is denoted as C[a,b] and is a subset of the set of all functions defined over the same interval.

What is the importance of the "Space of continuous functions C[a,b]"?

The space of continuous functions C[a,b] is important because it allows us to define and study functions that have a continuous behavior over a specific interval. This is useful in many areas of mathematics and science, such as calculus, differential equations, and physics, where continuous functions are often used to model real-world phenomena.

How is the "Space of continuous functions C[a,b]" different from other function spaces?

The space of continuous functions C[a,b] is different from other function spaces, such as the space of differentiable functions or the space of integrable functions, because it only includes functions that are continuous over the entire interval [a,b]. Other function spaces may include functions with different properties, such as being differentiable or integrable, but not necessarily continuous.

What are some properties of the "Space of continuous functions C[a,b]"?

The space of continuous functions C[a,b] has several important properties, including being a vector space, meaning that it is closed under addition and scalar multiplication. It is also a metric space, meaning that it has a distance function defined on it. Additionally, it is a complete space, meaning that every Cauchy sequence of functions in C[a,b] converges to a function in C[a,b].

How is the "Space of continuous functions C[a,b]" used in real-world applications?

The space of continuous functions C[a,b] has many real-world applications, particularly in physics and engineering. For example, it is used to model the behavior of continuous systems, such as the movement of a particle or the flow of a fluid. It is also used in optimization problems, where the goal is to find the most efficient or optimal solution within a continuous set of possibilities.

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