Spaces of continuous functions and Wronskians

In summary, the conversation discusses continuous functions as subspaces of each other, and the correct notation for representing them. Examples of functions in different subspaces are also given. The concept of the Wronskian and its use in determining linear independence is explained. An example of two functions in C^1(-∞,∞) that are also in C^∞(-∞,∞) is provided to illustrate the idea. The conversation ends with a confirmation of understanding.
  • #1
TaliskerBA
26
0
I'm struggling to understand continuous functions as subspaces of each other. I use ⊆ to mean subspace below, is this the correct notation? I also tried to write some symbols in superscript but couldn't manage. Anyway I know that;

Pn ⊆ C∞(-∞,∞) ⊆ Cm(-∞,∞) ⊆ C1(-∞,∞) ⊆ C(-∞,∞) ⊆ F(-∞,∞)

I know the axioms required to be a polynomial, but I am struggling to conceive the difference between, say, C(-∞,∞), C1(-∞,∞) and C2(-∞,∞) for example. Could someone possibly write out a few examples of functions that would, for example be vectors of C(-∞,∞) but not of C1(-∞,∞). I really appreciate any light that can be shed on this as I have been struggling to get it for a while.

Also when it comes to the Wronskian, I know how to use it to show linear independency, but I don't actually understand why. Specifically, why are the derivatives of functions important in determining whether a homogenous linear combination of functions has only the nontrivial solution, because I was under the impression that because the system is linear, derivatives don't come into it . I always prefer to understand the maths I am applying, so please help me!

I really appreciate any help.
 
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  • #2
Asking the difference between C and C^1 just boils down to finding a continuous function that is not differentiable. Surely you know of such a function. The most memorable example is probably the absolute value function x-->|x|.

A function that's in the difference C^1 - C^2 is x^{3/2}. More generally, x^{(2k+1)/2} is in C^k but not C^{k+1}.
 
  • #4
Sorry if I'm being slow, but how is x^(3/2) continuous for all values of x? Surely it is only continuous for (0,∞). Or does this matter? Furthermore I can find the first and second derivative of this with:
f'(x) = (3/2)x^(1/2)
and
f''(x)=(3/4)x(-(1/2))
and they are both continuous for (0,∞) but not for C(-∞,∞) (aren't they?)
 
  • #5
You're right. Make that |x|^{3/2}.

This one is well defined on all R and is differentiable at 0 because
[tex]\lim_{h\rightarrow 0}\frac{|h|^{3/2}-|0|}{h}=\lim_{h\rightarrow 0}\sqrt{h}=0[/tex]
 
  • #6
OK, I think the penny might have dropped, but if I write out my understanding of things please can you reply letting me know if I have got the jist (this ties back into the wronskian I was mentioning earlier).

Say I have f1=cosx and f2=sinx

The wronskian tells us that the determinant of (forgive my inability to write out a proper matrix):

cos x sinx
-sinx cos x

is 1. Therefore since this does not equal zero we know the two functions form a linearly independent set in C^1(-∞,∞). However, in truth this linearly independent set is also in C^∞(-∞,∞) since every fourth derivative of cosx = cosx, and the same goes for sinx therefore you can repeat the process of finding the derivatives an infinite number of times and they will always be continuous.

Have I got this right?
 
  • #7
Yes, completely right.
 
  • #8
Brilliant, thankyou!
 

1. What is a space of continuous functions?

A space of continuous functions refers to a set of all possible continuous functions on a given interval or domain. It is denoted by C(X), where X represents the interval or domain.

2. How is the continuity of a function determined?

A function is considered continuous if it has no sudden jumps or breaks in its graph. Mathematically, this means that the limit of the function as x approaches a particular point must be equal to the value of the function at that point.

3. What is the purpose of studying spaces of continuous functions?

Studying spaces of continuous functions is essential in many fields of mathematics, such as analysis, topology, and differential equations. It allows us to understand the properties of continuous functions and their behavior on a given interval, which can be applied in various real-world problems.

4. What is a Wronskian?

A Wronskian is a mathematical concept used to determine the linear independence of a set of functions. It is denoted by W(f1, f2,..., fn) and can be calculated using a determinant involving the derivatives of the functions in the set. If the Wronskian is non-zero, the functions are linearly independent.

5. How are Wronskians related to spaces of continuous functions?

Wronskians are closely related to spaces of continuous functions because they are used to determine the linear independence of a set of functions within a given interval. This is important in understanding the structure and properties of a space of continuous functions.

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