# 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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### (uniformly continuous) is it true?

Suppose $f$ is continuous on $(0,\infty)$ and $f$ is simmilar $g$ for all $x>M$ ($M>0$). (i.e. far any $\epsilon >0$, there is $M>0$ such that if $x>M$, then $|f-g|< \epsilon$) is ...
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### Is $f(x)$ constant under these conditions?

Statement Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be an function that is concave up and increasing. If $\displaystyle \lim_{x\to \infty}\frac{f(x)}{x}=0$, then $f$ is constant. It'll be easy if ...
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### Let $D'$ be the set of all accumulation points of $D\subset \mathbb{R}$. Show that $D\cup D'$ is closed.

I am trying to prove: Let D ⊂ R and D′ the set of all accumulation points of D. Let D = D ∪ D′. Show that D is closed. I am very confused and unsure of how to do this. I would appreciate it if ...
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### Why does the norm of a linear functional $T$ satisfy $\|T\|_*=\sup\{|T(f)|\mid f \in X, \|f\|\leq 1\}$?

For a normed linear space $X$, a linear functional on $X$ is said to be bounded provided there is an $M \geq 0$ for which $|T(f)|\leq M\|f\|$ for all $f \in X$. Denote $\|T\|_*$ the infinmum of all ...
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### Is it natural how $L^p$ spaces measure local and global sizes the same?

This is a continuation of my question Spaces of functions similar to $L^p$ but with different local and global sizes. I have been bothered by the fact that the $L^p$ norm on $\mathbb R^n$, which is ...
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### Is this proof correctly written? Show that the sum of two uniformly continuous functions on $A$ is uniformly continuous on $A$

If $f$ and $g$ are uniformly continuous functions in $A$ show that $f+g$ is uniformly continuous in $A$. Proof: because $f$ and $g$ are uniformly continuous on $A$ we can write ...
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### Saturation of a measure Folland Problem 1.3.16

Exercise 16 - Let $(X,M,\mu)$ be a measure space. A set $E\subset X$ is called locally measurable if $E\cap A\in M$ for all $A\in M$ such that $\mu(A) < \infty$. Let $\tilde{M}$ be the ...
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### Assume f and g are defined on all of $\mathbb{R}$ and that $\lim_{x\to p} f(x) = q$ and $\lim_{x\to q} g(x) = r$.

(a) Give an example to show that it may not be true that $\lim_{x\to p} g(f(x)) = r$ If we are to assume that f and g are defined on all of $\mathbb{R}$, wouldn't that mean that f and g are ...
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### Class of the derivative of a bilinear map

This question is more conceptual than other thing. We know that if $f:U\subset \mathbb{R}^n \to \mathbb{R}^m$, where $U$ is a open subset of $\mathbb{R}^n$, is differentiable, then the derivative of ...
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### Show that $\sup(\mathbb{Q} \cap (a,b)) = b$

Let $a<b$ be two real numbers. Show that $\sup(\mathbb{Q} \cap (a,b)) = b$ and $\inf(\mathbb{Q} \cap (a,b)) = a$. This intuitively makes sense. Since a sequence of rationals will infinitely ...
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### Uniformly bound exists for a continuous function sequence in a neighbhorhood of a convergent point?

Assume that $$\lim_{n\to\infty}f_n(x_0) < \infty$$ and also that $\forall n\ge1$, $f_n(x)$ is continuous in a neighborhood of $x_0$, say $(x_0-\delta_1, x_0+\delta_1)$. Besides, for any fixed $x$ ...
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### What are vector norms used for?

I'm currently working with a computer science problem that requires me to build vectors that can return their own norms. Based on Wolfram Alpha's description, I think I have an idea of how this is ...
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### Unit ball of $L^1$, $L^\infty$ and $C(X)$ is not strictly convex

I need to show that the unit balls of $L^1(\mu)$, $L^\infty(\mu)$ and $C(X)$ are not strictly convex. I have already shown that if $1<p<\infty$ then the unit ball of $L^p(\mu)$ is strictly ...
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### Is being real analytic at a point equivalent of matching the Taylor Series around that point?

Let $U \subset \mathbb{R}$ be an open interval and let $x_0 \in U$. Let $f: U \to \mathbb{R}$ be defined by a power series around $x_0$ with radius of convergence $R > 0$, f(x) = ...
### Showing any bounded sequence in Holder space $C^{1/2}$ has a convergent subsequence in Holder space $C^{1/3}.$
Prove that any bounded sequence in $C^{1/2}([0,1])$ admits a convergent subsequence in $C^{1/3}([0,1]),$ where we say that $f \in C^{\alpha}([0,1])$ if $f$ is Holder continuous of order $\alpha.$ The ...
By Cauchy formula, the radius of convergence of the series $\sum_{n=0}^{\infty}a_nr^{n}$ is $\rho=1/\limsup\limits_{n\rightarrow +\infty}\sqrt[n]{|a_n|}$. Let $\{\lambda_n\}_{n=0}^{\infty}$ be an ...