Lucas Zanella

I've just arrived now at $(\frac{1}{\sqrt{4\pi t}}\int|\partial_x^n e^{-x^2}|^2)^{1/2}$ which is almost $||\partial_x^n(x)\phi_{1/4}||$, but I don't get how $C^{n+1}$ appears

$\phi_{1/4} = e^{-x^2}/\sqrt{\pi}$ right?

Take a look at my update, I don't know what you said about canceling for every $n$ but if I do the rule $\int_{\mathbb R} f(\lambda x)^2 dx = \frac1\lambda \int_{\mathbb R} f^2(y) dy$ I get $\phi_{1/4}$

I arrived at $\|\partial^n_x\phi_t\|_{L^2} = \|\partial_x^n \phi_{1/4}\|_{L^2}$ does that mean I can simply say this is less than $C^{n+1}\|\partial_x^n \phi_{1/4}\|_{L^2}$ for some $C^{n+1}$ greater than $1$?

How does $\pi$ appears? I think I understood it terribly wrong, because the first derivative with respect to $x$ is $-\frac{2x}{4t}e^{-\frac{x^2}{4t}}$

I finally understood all norms, but how did you do that Chain Rule? I can't reproduce here where the $\sqrt{}$ appears and where $\phi_{1/4}$ appears (I know what $\phi$ is)

I was in doubt because $t$ can also vary so I could take the norm in $t$, but apparently you made in $x$

From line $4$ to $5$ you go from $L_y^2$ to $L^2$

Thanks, I understoodd. But now what does $L^2$ alone mean? In which variable?

what do you mean when you write $L_x^2$ for example? What is the sub $x$ for?

Can you be more specific in the norm of the derivative of a convolution and Cauchy Schwarz being applied to it? I don't understand how these $L^2$ norms appear, I think I don't have background on these things

@DanielFischer that's what I needed. Thank you.

@DanielFischer thanks! Steinitz exchange lemma says: "If {v1, ..., vm} is a set of m linearly independent vectors in a vector space V, and {w1, ..., wn} span V then m ≤ n and, possibly after reordering the wi, the set {v1, ..., vm, wm + 1, ..., wn} spans V."

@JohnDoe but Gram-schimdt is for creating new basis. I need the theorem that allows you to complete a set to a basis

Please, somebody knows what is the name of the theorem that allows you to complete a set of linearly independent vectors into a basis? I only know its name in portuguese. Need a wikipedia page for it.

Exactly! Thank you so much @AntonioVargas

The bar gets tiny

sorry

tiny

@AntonioVargas I tried but the bar is too litle

How to write that vertical bar when I want to say that the antiderivative goes from a to b, for example?

Ok, thanks by all, see you guys another time!

LaTeX is not accepted here?

$\frac{1}{2}$ testing LaTeX

@mixedmath Thank you so much, again, now it works

:D

I'm gonna log in again

@mixedmath Thank you so much :)

@mixedmath yes!

@mixedmath my account is: math.stackexchange.com/users/55643/lucas-zanella

@mixedmath for months!

How do I change my name on this chat? My stackexchange does not have this name :c

Test

font: angelustenebrae.livejournal.com/15908.html#

@PedroTamaroff right

math is nice