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Jul
15
comment What if we removed all the irrational numbers from the real number line?
You might find this enlightening: dpmms.cam.ac.uk/~wtg10/mathsindex.html. Especially the section on analysis.
Jul
10
revised Span of Dirac's delta distributions dense in Hilbert space of $L^2$ functions?
Use \langle and \rangle (or their unicode equivalents) instead of < and >
Jul
9
comment The binomial formula and the value of 0^0
But the use of $\sum_{i} |x_i|^0$ as the definition of the zero "norm" seems to me to have basically nothing to do with exponentiation since it's actually computing the Hamming distance to $0$. And if one argues that it can be realised as the limit of $p$-norms as $p$ goes to zero, then continuity and limits are coming back in to the picture and it's no surprise that all bets are off.
Jul
8
comment Basis in the vector space of all polynomials
Furthermore, the Fredholm alternative doesn't apply in the case of $\mathbb R[x] \cong c_c$ since it's not complete.
Jul
7
comment Every uncountable subset of $\mathbb{R}$ has a limit point
Here's a much simpler argument: There must be an $n$ such that $A := E \cap [-n,n]$ is infinite. $A$ is an infinite subset of the compact set $[-n,n]$ so it has a limit point in $[-n,n]$. Now argue that such a limit point must also be a limit point of $E$.
Jul
2
comment Why do determinants have their particular form?
One thing that one has to verify regarding $T(\bigwedge_{i=1}^n v_i) := \bigwedge_{i=1}^n T(v_i)$ is that there could be different vectors $w_1,\dotsc,w_n$ such that $\bigwedge_{i=1}^n v_i = \bigwedge_{i=1}^n w_i$, so you have to check that you get the same result on the right hand side regardless of the choices you make.
Jul
2
comment Why do determinants have their particular form?
The multilinearity and alternatedness of the determinant suggests a connection with the exterior algebra of the vector space in question. And indeed there is such a connection (q.v. the linked article on Wikipedia).
Jul
2
awarded  Curious
Jun
29
comment Topology of convergence in measure
@Adam $L^0$ usually denotes the set of measurable functions. I don't think the topology it's usually given is normable (Like $L^p$ for $0 < p < 1$ isn't). en.wikipedia.org/wiki/Convergence_in_measure#Topology
Jun
23
comment Is the arbitrary union of consecutively non-disjoint path-connected sets path-connected?
@EricStucky the problem is that any point $(\alpha,a)$ has uncountably many copies of $[0,1)$ between it and $\infty$.
Jun
23
comment Is the arbitrary union of consecutively non-disjoint path-connected sets path-connected?
@EricStucky after some thinking it seems that the extended long line might not work. The point at infinity isn't connected by a path to anything else.
Jun
19
comment What are some motivating examples of exotic metrizable spaces
$B_1$ is also an excellent example of an open ball with many centres: $B_1 = B(a,1-a)$ for any $0 \leq a < \tfrac12$. Although it's not as extreme as the situation in an ultrametric space
Jun
18
revised Riesz Lemma for reflexive spaces
edited body
Jun
18
comment Let $T: \Bbb R^3 → \Bbb R^2 $be a linear transformation defined by $T( x, y, z) = ( x + y, x - z)$
LaTeX tips: Do not use \Bbb it's a mathjax thing and won't work in actual LaTeX files. Use \mathbb instead. E.g. \mathbb R: $\mathbb R$. You should use \dim to get $\dim$. It's useful to use abbreviations like \ker T ($\ker T$) and \operatorname{im} T ($\operatorname{im} T$). Yes, LaTeX and mathjax don't come with a predefined \im and \Im gives the symbol for imaginary part of a complex number.
Jun
16
revised Why in this case $f$ should be entire?
added 20 characters in body
Jun
14
answered Is there any false case for that: $\exists x \in D, \forall y \in D, P(x, y) \implies P(y, x)$?
Jun
14
answered Can $\mu(A) < \liminf_{n\to\infty} \mu (A_n)$?
Jun
14
revised Can $\mu(A) < \liminf_{n\to\infty} \mu (A_n)$?
edited body
Jun
14
comment Can $\mu(A) < \liminf_{n\to\infty} \mu (A_n)$?
Further TeX tips: Use \liminf and do not overuse \left .. \right. Most of the time they're unnecessary and sometimes they make things look wrong.
Jun
14
revised Can $\mu(A) < \liminf_{n\to\infty} \mu (A_n)$?
edited title