# Tagged Questions

Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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### Riesz's 1909 proof of the Riesz Representation Theorem

Frigyes Riesz originally proved the Riesz Representation Theorem on $C[0,1]$ -- here is his 1909 paper in English (original French). He builds a real valued function $\text{A}$ on $[0,1]$ ...
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### Lagrange multipliers in Calculus of Variations

I am trying to learn about Calculus of Variations and I am beginning to see some constrained optimization problems in the domain of functionals, by using Lagrange multipliers. It seems that things ...
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### A sequence of functions that is uniformly continuous, pointwise equicontinuous, but not uniformly equicontinuous when their domain is noncompact

I'm trying to prove my sequence of functions $(f_n) = \frac{n}{n+1}\cos(x^2)$ on (0,1) is pointwise equicontinuous, uniformly continuous, but not uniformly equicontinuous. But, I'm having a lot of ...
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### Another way of expressing $\sum_{k=0}^{n} (-1)^k\frac{H_{k+1}}{n-k+1}$

In this post Another way of expressing $\sum_{k=0}^{n} \frac{H_{k+1}}{n-k+1}$ I asked for a solution of the non-alternating series. How about the alternating series? Can we find a nice way of ...
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### Could $4+2+4+2+4+2+\cdots = -1$?

In physics classes, on this StackExchange and even in blogs the sum $1 + 2 + 3 + 4 + \cdots = - \frac{1}{12}$ has been under the microscope. Why does $1+2+3+\dots = {-1\over 12}$? The Euler-...
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### How to find a homeomorphism $\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ having certain properties

Let $n\ge 2$ and let $C$ be a Cantor space in $\mathbb{R}^{n}$. That is, $C$ is homeomorphic to the Cantor ternary set. Let $x$ and $y$ be two points in $\mathbb{R}^{n}-C$, and let $L_{xy}$ be the ...
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### Equality condition in Minkowski's inequality for $L^{\infty}$

I am trying to find out when equality holds in Minkowski's inequality for $L^{\infty}$ (i.e. a necessary and sufficient condition for equality). I did a search and there was a discussion for the case ...
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### Unit Ball in $C([a,b])$ is not Compact

Show that the assumptions of the Heine-Borel theorem do not $\implies$ compactness in $C^0([a,b])$ ( Hint, consider the set of functions $S=\{f\in C^0([a,b]): f \ \text{is uniformly bounded by} \ 1\}$ ...
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### Capacity vs measure of a set - intuitive understanding

There is a concept of measure of "largeness" of a set, called capacity. The intuition is, instead of physical largeness (measured by Hausdorff or Lebesgue measure), capacity measures how good a given ...
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### How do people pick $\delta$ so fast in $\epsilon-\delta$ proofs

For example, in a proof that shows $f(x) = \sqrt(x)$ is uniformly continuous on the positive real line, the proof goes like: Let $\epsilon > 0$ be given, and $\delta = \epsilon^2$.... Or to ...
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### boundness of linear map

Let $X$ and $Y$ be Banach spaces, $T : X\rightarrow Y$ a linear map such that $f \circ T\in X^*$ for every $f\in Y^*$, then $T$ is bounded. I've tried to show that $T$ is closed, but I think that's ...
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### Estimation of a sequence related to the Stirling's formula

I need to show that $$n!=\left(\frac{n}{e}\right)^n\sqrt{2\pi n}e^{\lambda_n}$$ where $$\frac{1}{12(n+1)}<\lambda_n$$ I calculated that $$\lambda_n=\ln n!+n-n\ln n -\frac{1}{2}\ln(2\pi n)$$ On ...
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### What is a distribution in $H^{-1}(\Omega)$?

Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ with $\partial \Omega$ being $C^2$. Suppose $u\in H^1_0(\Omega)$, $f\in L^2(\Omega)$ and $\nu>0$. It is said in the Navier-Stokes Equations ...
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### Is there a continuous nowhere Lanczos generalized differentiable function?

The Lanczos generalized derivative comes from local regression, and is described in this pdf. (You do not need to read that to understand my questions.) If you do read it, observe that I'm pulling ...
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### Can we use sequences to test continuity of a weak$^*$-continuous operator?

Let $X,Y$ be Banach spaces. Now assume we have a map $T:X'\rightarrow Y'$ where $X'$ and $Y'$ are equipped with the weak$^*$ topology and not the norm topology. Can I infer from this that an operator ...
If $f:[0,1]\rightarrow \mathbb{R}^d$ is a smooth curve, then is there are relationship between the total variation of $f$ and the geodesic curvature of $f$? I expect they both should be zero iff $f$ ...