let C0 be the set of continuous functions f : R -> R. For n >= 1, let Cn denote theset of functions f : R -> R such that f is differentiable and such that f' is contained in C(n-1). (Therefore Cn is the set of functions whose derivatives f',f'',f''',....,f^(n) up to the nth order exist and are continuous.) Prove by induction that Cn is a subspace of V where V is the set of all functions f : R -> R.(adsbygoogle = window.adsbygoogle || []).push({});

There are three properties that Cn must satisfy to be a subspace,

1.) it must contain the zero vector of V

2.) It must be closed under vector addition

3.) it must be closed under scalar multiplication

I am not sure which of these properties i must perform induction on (obviously not 1.) ) or should it be both 2.) and 3.)..?

I would greatly appreciate it if someone could give me a hint for what the inductive step should be..?

cheers,

James

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# Proving a set of derivatives to be a subset of real functions

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