Hey /sci/, if you take a sequence of functions...

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Hey /sci/, if you take a sequence of functions such as

[math]f_n(t) = a_n cos(n.t) + b_n sin(n.t)[/math] such as f_n tend to the zero function, show that a_n and b_n tend to 0

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>>7840368

That's not homework.

Just interested on how you do it.

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What does "f_n tend to zero" mean ?

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>>7840388

probably that f_n converges to 0

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>>7840392

I know that, but on an infinite dimensional space, there are many nonequivalent ways a sequence can converge

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>>7840371

>>7840370

>click link out of sheer ignorance

>first image is two hairy men on their backs going ass to ass with a dildo

I've been here since 2007 if you can believe that.

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>>7840393

pointwise convergence

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>>7840393

This is false, unless you mean under different topologies. I'm pretty sure function spaces have a canonical topology, though.

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>>7840602

Well [math]f_{2n}(\pi) = a_n[/math] and [math]f_{4n+1}\left(\frac{\pi}{2}\right) = b_n[/math]

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>>7840748

Yes, of course, I meant different topologies (although if you set non-Hausdorff, sequences might have several limits but that was not my point).

Also, no, function spaces don't have "canonical topologies". For many reasons, there are topologies that are natural to consider. For example, on [math]C^1([0,1])[/math], a very natural topology to consider is the one associated with the "C1"-norm [math]N: f \mapsto ||f||_{\infty} + ||f'||_{\infty}[/math]. It is natural because it makes it into a complete space (more heuristically, it makes use of everything we know: the natural convergence on bounded continuous functions is uniform so we use the uniform norm of f and of f'). But we could have also considered sobolev norms [math]||\cdot ||_{1,p}: f \mapsto ||f||_{L^p}+||f'||_{L^p}[/math] or the weak topology associated with N, or the pointwise convergence topology, or the uniform norm [math]||\cdot ||_{\infty}[/math].

All of these have use in a context or another and knowing how to juggle between them is important.

Also, there are situations where finding even one interesting topology is a challenge: For example, it is not obvious how to set an "interesting" topology on [math]C^{\infty}(\mathbb R)[/math] (it inherits all the C^k norms but none of these can individually capture all the information)

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