Proving the Nested Compact Set Property
There’s this theorem in my analysis book that I want to prove, which states that:If$k_{1} \subseteq k_{2} \subseteq k_{3} \subseteq ...$Then the intersection$\bigcap_{n=1}^{\infty}k_{n} \neq...
View ArticleLinear extension of a functional on closed unit ball.
Let $X$ be a normed linear space with closed unit ball $B$. Suppose the function $f: B\rightarrow [-1,1]$ has the property that whenever $x,y,x+y$ and $\lambda x$ belong to $B$, $f(x+y)=f(x)+f(y)$ and...
View ArticleAssociativity of Convolutions
In Folland's real analysis textbook, there are the following propositions:Assuming that all integrals in question exist, we have$$(f*g)*h=f*(g*h) $$The proof is based on the Fubini's theorem.But I...
View ArticleProve: If $f_1$ and $f_2$ are integrable, then $f_1\vee f_2$ is integrable...
I need to prove the following result:Let $(X,\mathscr{A},\mu)$ be a measure space. If $f_1$ and $f_2$ are integrable, then $f_1\vee f_2$ is integrable over each $A\in\mathscr{A}$.Here is my...
View ArticleSuppose $f:[a,b]\to \mathbb{R}$ is a continuous semi-differentiable function....
Let $f:[a,b] \to \mathbb{R}$ be a continuous function on $[a,b]$ such that there exists $f_{+}'(x) \in \mathbb{R}, (\forall) x \in [a,b)$. Is it true that there exists a countable set $D \subset [a,b]$...
View ArticlePointwise convergence of $f_{n}(x) = \sin(\frac{x}{n})$
I'm trying to prove wether $f_{n}(x) = \sin(\frac{x}{n}), f_{n}:\mathbb{R}\to\mathbb{R}$ has pointwise convergence or not. My first idea is it is pointwise convergent to the function $f = 0$ because...
View ArticleDifferentiable Functions / Zorich
A body that can be regarded as a point mass is sliding down a smooth hill under the influence of gravity. The hill is the graph of a differentiate function y = f(x).a) Find the horizontal and vertical...
View ArticleUniform convergence of $\{f_n\}$ satisfying $f_n\left(x + \frac{1}{n}\right)...
I encountered this problem on a graduate school entrance test :Let $\{f_n\}$ be a sequence of real-valued continuous functions on $\mathbb{R}$ such that $$f_n\left(x + \frac{1}{n}\right) = f_n(x)...
View ArticleCan a function be differentiable but not strongly differentiable (Knuth)?
Donald Knuth defined $f$ is strongly differentiable at$x$ if $$f(x+\epsilon) = f(x) + \epsilon f'(x) + \mathcal{O}(\epsilon^2)$$ for sufficiently small $\epsilon$.What differentiable functions are not...
View ArticleIs $r$ when $\lim_{r\rightarrow 0}$ "equivalent" to $\frac{1}{n}$ when...
The question I am trying to do is:Let $f:U\in\mathbb{R}^n\rightarrow\mathbb{R}$ be a continuous function.Show that for every $x\in U$ we have: $$f(x)=\lim_{r\rightarrow...
View Article$\sum 2^{-r_n}/r_n$ diverges $\implies$ $\sum 2^{-\lceil r_n \rceil} /...
I want to prove $\sum 2^{-r_n}/r_n$ diverges $\implies$$\sum 2^{-\lceil r_n \rceil} / {\lceil r_n \rceil}$ diverges where $r_n$ is a nondecreasing sequence of reals. This came up in Billingsley...
View ArticleProve that the set of injective linear transformations is an open set.
Prove that the set of injective linear transformations from $\mathbb{R}^n$ to $\mathbb{R}^m$ is an open set. Using the fact that a Linear transformation is injective if and only if there is...
View ArticleSolving for f when f(x+y)=f(x)+f(y)+axy where a is a real number
This question has been asked before here but I have followed a different approachWe have:$$f(0) = 0$$$$f(x) = f(\frac{x}{2}+\frac{x}{2}) = 2f(\frac{x}{2})+a\frac{x^2}{4}$$$$f(\frac{x}{2}) =...
View ArticleRegarding relative compactness in baby rudin
I am an early undergraduate student, self-studying baby rudin.I have encountered a problem in Thm. $2.33$, it states that:Suppose $K \subset Y \subset X$. Then $K$ is compact relative to $X$ if andonly...
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Orbit of vector field crosses transverse section in the same direction
Let $X\in\mathbf{C}^1(U,\mathbb{R}^2)$ a vector field on the open set $U\subset\mathbb{R}^2$. Let $D\subset\mathbb{R}$ open and $f:D\rightarrow U$ be a $\mathbf{C}^1$ map such that $\{f'(x),X_{f(x)}\}$...
View ArticleIt there a better upper bound for $\max_{1\leq i \leq n}(a_{i}*b_{i})$?
I would like to give an upper bound for $\max_{1\leq i \leq n}(a_{i}*b_{i})$ which is related to $\max_{1\leq i \leq n}(a_{i})$. I know that $\max_{1\leq i \leq n}(a_{i})\max_{1\leq i \leq n}(b_{i})$...
View ArticleFractional part of a sum
Define for $n\in\mathbb{N}$$$S_n=\sum_{r=0}^{n}\binom{n}{r}^2\left(\sum_{k=1}^{n+r}\frac{1}{k^5}\right)$$I need to find $\{S_n\}$ for $n$ large where $\{x\}$ denotes the fractional part of...
View ArticleCantor's Isomorphism Theorem for Countable Dense Subsets of $(0,1)$
By Cantor's Isomorphism Theorem, we know that any two given countable dense subsets $D_{1}, D_{2} \subseteq (0,1)$ are order isomorphic. However, this order isomorphism may be quite irregular in the...
View Articlehow to find explicit formula for $f_{n+1}=af_n+\frac{b}{f_n}$?
I tried to find an explicit formula for the recurrence relation$$f_{n+1}=af_n+\frac{b}{f_n} , f_0 \ne0$$I will show what I got in five casesCase1for $f_0=-\frac{\sqrt{b}}{\sqrt{1-a}}\ne0$ I got that...
View ArticleIs the Evaluation Map on Bounded Borel Measurable Functions Borel Measurable?
I am working with a set $I$, defined as the closed interval $[0,1]$, and a set $X$, which consists of all bounded Borel measurable functions defined on $[0,1]$. The uniform metric on $X$ is defined...
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