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## Main Question or Discussion Point

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.

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.