I'm trying to show that the space $C[0,1]$ with the sup metric and $L^2$ metric are not equivalent in the sense of topological equivalence: en.wikipedia.org/wiki/Equivalence_of_metrics
I would scale the functions in such a way that their $L^2$ norm is $1$, i.e. consider $g_n = f_n / \|f_n\|_2$. Then you can see that the $\|\cdot\|_\infty$ norm of $g_n$ tends to infinity.
@tb Hmm, no... What do you mean? You told me you started doing GGT. So, I had that in mind under the hypothesis that you're doing this stuff only now or something of that sort!
@BenjaminLim because I find it more intuitive to say that the closed $\|\cdot\|_2$-unit ball contains elements of arbitrarily large $\|\cdot\|_\infty$-norm.
So it cannot be contained in any $\|\cdot\|_\infty$ ball
@BenjaminLim I'm not quite sure what you're saying. We know that every $\|\cdot\|_2$-ball contains an $\|\cdot\|_\infty$-ball, so the $\|\cdot\|_\infty$-topology is finer. Now we produced an example showing that there can't be an inclusion in the other direction, so the topology is strictly finer.
@tb In australia labour is very expensive so most of the time people fix their own stuff. For example at my uncle's house I have to help him dig the garden and stuff.
I don't find it that boring. I find it more inconvenient to wait one morning for some guy to come to do something I could do myself in less than 10 minutes.
@BenjaminLim that was more or less my point. I don't do either of them seriously enough to say I am working in those fields but what I'm doing has to do with them...
Less than 10 minutes? Heh. To me it sounds as if you have to remove the mirrored cupboard thingie (that alone takes 10 minutes) and then fiddle with cables.
I'm trying to plot the phase of this signal $s(f)=A^2T^2sinc^2(Tf)e^{-(j\pi Tf)}$
my problems came when I add the delay.I understand how to plot it without delay but adding the delay I didn't understand how plot.some help?
@MattN no, I have to remove the lamp shade: that takes maybe half a minute. Then I have to undo two screws and take something off, one more minute. Then I have to take a cable with three wires and plug those into six prepared thingies (which the guy left there, of course). Then I have to re-install the thingie, fix two screws and put the shade back on. Ten minutes are amply sufficient for that. :)
@MattN well, because you chose to use covering spaces, which are an unnecessary diversion. You can just use the functional equation of the exponential function.
@JM I'd guess that depends on your skill : ) I've not tried. I used to practice in a park to avoid disturbing my house mates but I'm also shy so I did that at 6 am when there weren't that many people in the park.
@tb: (1) I wonder where I can find that version of Riez representation theorem Didier used?
(2) In Didier's reply, is the weak convergence in $ L^p, p \in [1, \infty]$ wrt $L^q, 1/p+1/q=1$, both weak convergence and weak* convergence? Because $L^p$ and $L^q$ are continuous dual to each other?
@Tim (1) you just have to observe that a probability measure gives a continuous functional of norm one on $C_b(X)$ and that two distinct measures give rise to two distinct functionals (an exercise you should do!). (2) $L^p$ is reflexive for $1 \lt p \lt \infty$. For $p = 1,\infty$ this is wrong. Try to come up with counterexamples.
I'm trying to plot the phase of this signal $s(f)=A^2T^2sinc^2(Tf)e^{-(j\pi Tf)}$
my problems came when I add the delay.I understand how to plot it without delay but adding the delay I didn't understand how plot.some help?
