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16m
answered Proving existence of a linear functional
11h
awarded  Revival
11h
answered Difference between i and -i
11h
comment Proving existence of a linear functional
To separate sets $A$ and $B$ from each other, how about separating the difference set $A - B$ from the singleton $\{0\}$? That is a set-up for the Hahn-Banach theorem.
20h
comment Why can't the definition of convergence be alterted to this one?
You should try to find an example of a non-convergent sequence that satisfies that condition. (Presumably this is what the problem is asking for.)
1d
comment Alternate proof of a result on dual spaces: what is wrong with it?
If $g \not\in L^q$, then for some $f \in L^p$, the integral $\int_X fg \,d\mu$ does not exist. The functional $\Phi(f) = \int_X fg \,d\mu$ is unbounded (and therefore discontinuous) on its domain.
1d
comment Finding the PDF from the CDF where the CDF is not differentiable at some point
The PDF is defined only up to a set of measure zero. So one point is no problem. Indeed, a null set where it is undefined is also no problem. (Or course the CDF must be absolutely continuous, or there is no PDF at all.)
2d
comment Approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets?
Example: prove the Cauchy-Schwarz inequality. Prove the finite-dimensional case, then take a limit.
2d
comment Approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets?
One common way to deal with Hilbert space ... first prove your result in finite-dimensional space, in such a way that you can take a limit to get the Hilbert-space result.
2d
answered Extended Socratic Syllogisms?
2d
comment Almost surely vs expectation
Also, in probability theory the term "characteristic function" has a different meaning.
2d
comment Almost surely vs expectation
$X_i$ has value $i$ with probability $1/i$, otherwise zero. In probability theory we do not use $\chi$ like this, instead $1_{(0,1/i)}$.
2d
comment Non-standard models for Peano Axioms
It is important that we are using a first-order version of Peano's axioms. (Peano himself phrased induction as a second-order property, and Goedel need not apply in that case.)
2d
comment Calculation of $\lim\limits_{x \to 0} \frac{\frac{\mathrm d}{\mathrm d x} (e^{\sec x})}{\frac{\mathrm d}{\mathrm d x} (e^{\sec 2x})}$
Maple gets the answer $1/4$.
Dec
18
answered Baire sets in locally compact Hausdorff spaces
Dec
17
comment Baire sets in locally compact Hausdorff spaces
So there will be two parts to the proof. (a) $\mathcal A := \{B \cap F \; | \; B \in \mathcal Ba(X)\}$ is a sigma-algebra in $F$; and (b) every compact $G_\delta$ of $F$ belongs to $\mathcal A$.
Dec
16
answered If $f$ is continuous at $x_0$, then $|f|$ is continuous at $x_0$
Dec
16
answered Is this an identity: $\int_X f \, d\mu=\int_{\mathbb{R}} \mu(f^{-1}(t)) \, dt$?
Dec
16
comment Evaluate $\int_1^\infty \frac {dx}{x^3+1}$
Generally, contour integration is helpful to evaluate a definite integral if the integral is evaluated between two singularities of the integrand (or at least points special in some way for the integrand). Here, $1$ is not a special point for the integrand. Which means, perhaps: evaluating using a contour is the same work as evaluating without using a contour.
Dec
15
comment How to integrate using known distributions
If you know the moments of the Rayleigh distribution, that will do it.