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May
23
awarded  real-analysis
May
17
comment Proof of Cauchy Schwartz inquality from Terry Tao's notes doubt
The map is only $v\mapsto ve^{i\theta}$. If you apply this rotation to both $u$ and $v$, then you preserve $\langle u,v\rangle$, which is not what you want. Professor Tao's point is that in the inequality $|\text{Re}\langle u,v\rangle| \leq \frac{1}{2}(\|u\|^2 + \|v\|^2)$, there is an imbalance of rotation symmetry, and consequently there is a little wiggle room on the non-rotation-invariant LHS. Therefore the idea is to find a rotation that does not preserve the LHS, and $v\mapsto ve^{i\theta}$ is enough.
May
15
answered Proof of Cauchy Schwartz inquality from Terry Tao's notes doubt
May
1
revised Inverting a map from a finite 3D grid to 1D
removed [computational-complexity] tag
May
1
reviewed Approve Physical significance and graphical point of view of second derivative of a function $f''(x)$ .
May
1
reviewed Approve Calculus limits with sin and cos
May
1
comment function constant on arc is constant on boundary
@MichaelHardy I don't see any of your comments on that question...is there a reason you're asking me in particular? This seems rather random.
Apr
30
reviewed Close Proving that $\sum \limits_{d^2|n}\ \mu (d)=\mu^2(n)$, where $\mu$ is the Möbius function.
Apr
30
reviewed Close Error function etymology: Why the name?
Apr
30
reviewed Leave Open The difference between the ring version and module version of Chinese Remainder Thereom.
Apr
30
reviewed Leave Open Properties of Infinite Limits
Apr
30
reviewed Leave Open Verification of solution of a contest problem with a limit of nested radicals
Apr
30
reviewed Leave Open Calculating $a_n$ in $\sum_{n=1}^\infty a_n \sin(\frac{n \pi}{2})=T_0$
Apr
30
comment Can we estimate the dimension of some function space?
A paracompact Hausdorff space is normal, so Urysohn's lemma applies. Use it on a singleton and the complement of the interior of a compact neighborhood (if a larger one exists) to get a nontrivial function of compact support. (I'm not being exact here, since I don't know the full answer either, but I think this might be a decent starting direction.)
Apr
30
comment Can we estimate the dimension of some function space?
How about "if $X$ is nonempty," since if $X$ is paracompact Hausdorff then using Urysohn's lemma you can construct a nontrivial continuous function of compact support? And I think this extends easily to locally compact Hausdorff. Not sure about the non-zero linear functional part, though if $X$ is finite you'll get a finite-dimensional $C_c(X)$ and a dual of the same dimension.
Apr
30
comment function constant on arc is constant on boundary
I don't think my argument is particularly complicated...but I do like yours, I hadn't thought of it. If you don't like my argument that's ok, but I suggest you post your proof as an answer rather than using it as a belittling comment on my own.
Apr
30
comment Representing functions as power series and finding $c_0,c_1,c_2$…
Do you know the power series expansion of $1/(1-y)$? Can you modify this somehow to cover the function you're given?
Apr
30
revised function constant on arc is constant on boundary
added 56 characters in body
Apr
30
answered function constant on arc is constant on boundary
Apr
30
comment Can we estimate the dimension of some function space?
Is it not enough to say when the dimensions are finite? In most cases the dimension of these spaces should be infinite, with finite discrete spaces being exceptions to this rule.