Show that the inverse of the linear operator...
Show that the operator $l^{\infty }\rightarrow l^{\infty } $ defined by $y=(\eta _{j})=Tx$, $\eta _{j}=\frac{\xi _{j}}{j}$, $x=\xi _{j}$ has inverse operator not bounded.I though by continuity,...
View ArticleProof all derivatives of $ \frac{1}{1+e^{-\tan(\pi x - \pi / 2) }} $ tend to...
Our professor asked us to find a function $ g : (0,1) \rightarrow \mathbb{R} $ that extends$ f : \mathbb{R} - (0,1) \rightarrow \mathbb{R} $ that is constantly $0$ for $x\leq0$ and constantly $1$...
View ArticleProof that for any interval (a,b) with a
Background: We are assuming that the elements of $\mathbb{R}\setminus\mathbb{Q}$ are irrational number. If $x$ is irrational and $r$ is rational then $y=x+r$ is irrational. Also, if $r\neq 0$ then $rx$...
View ArticleConvexity of $f(x) = x^TAx^Tx + C$
Under what conditions this function will be a convex function.$$f(x) = x^TAx^Tx + C$$A is negative definiteA is positive semi-definiteA can be either positive or negative semi-definiteA is diagonalMy...
View ArticleReal analysis, Infimum, Supremum [closed]
Let ∅ 6= A, B ⊆ R and α∈ R. Consider the following subsets:A + B := {a + b : a ∈ A, b ∈ B}Show that, if A and B are bounded above, then so is A + B and sup A + B = sup A + sup B.\
View ArticleIs it true that if $f:\mathbb{R} \to \mathbb{R}$ is a continuous function and...
I'm pretty sure this is false and that I should use Cantor's function (its extension) as a counterexample, but I don't know how.
View ArticleUniqueness of function from an integral transform.
Suppose that $f,g$ are positive, non-decreasing, differentiable functions, with $f(0)=g(0)=0$. Further suppose that $f,g$ are such that $c\sqrt{x}\leq f(x),g(x)\leq Cx$ for all large $x$, and such...
View Articleapplicability of Fubini-Tonelli theorem
I read on wikipedia (the only free source I could find that talks about the boundaries of integration) that the Fubini-Tonelli theorem can be applied only on double integrals that have rectangular...
View ArticleBounded variation + injective implies piecewise strictly monotonic almost...
The question is can we prove the following: Consider $a,b\in {\bf R}$, $a<b$.(1) If $u\in {\rm BV}([a,b])$, and(2) if $u$ is an injection on $[a,b]$,then there exists $u_0:[a,b]\rightarrow {\bf R}$...
View ArticleBounded variation + injective implies strictly monotonic?
The question is are the following assertions true:I.(strong version) Consider $a,b\in {\bf R}$, $a<b$.(1) If $u\in {\rm BV}([a,b])$, and(2) if $u$ is an injection on $[a,b]$,then $u$ is strictly...
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Inequality involving entropies: $\left \|p -\frac{1}{n} e \right \|_2\ge\left...
For a given probability vector $p=(p_1,\dots,p_n)$ with $p_1,\dots,p_n > 0, \sum_{i=1}^n p_i=1$ and with $e:= (1, \dots, 1)$, I want to prove the following inequality:$$\small\left \|p -\frac{1}{n}...
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Equivalent Sets of Order Axioms for the Real Numbers
I'm currently studying Königsberger Analysis 1, 6th Edition (in German) , which defines the Order Axioms for the real numbers like this:Translation from German :The Order of $\mathbb{R}$ is defined by...
View ArticleConstructing measurable sets out of the unit square
We know that for a Lebesgue measurable set $S \subseteq \mathbb{R}$, any of its translates $x+S$ for some $x \in \mathbb{R}$ is Lebesgue measurable with $m(x+S) = m(S)$. I am wondering about properties...
View Articlereal numbers in base 2 - how the infinite series (Σ(b/2^n)) converge to a...
Any real number in $[0,1]$ has a unique binary decimal representation 0.bbbbb, where each b is either $0$ or $1$. Numerically, $0.b_1 b_2 b_3 b_4 b_5...=\sum^{\infty}_{i=1}(b_n/2^n) $(***b_1=b-sub-one)...
View ArticleDecimal expression of reals
Let $x>0$ be real. Then $A_1=\{n\in \mathbb{N}\mid x<n\}$ is nonempty since $\mathbb{R}$ is dedekind complete. Since $\mathbb{N}$ is well ordered, $A_1$ has a least element $k$. Thus $k-1$ is the...
View ArticleWeierstrass... thing
There is in my maths text-book this property/theorem given under the name of Weierstrass property/theorem:Let $ (a_n) $ be a sequence of real numbers.a)If $ (a_n) $ is monotonically increasing and has...
View ArticleGiven $x >0$ and $n$ is contained in $N$, show that there exists a unique...
Also, $r$ is usually denoted by $x^{\frac{1}{n}}$I'm having a lot of issues with this problem. It is a challenge problem and I feel as though I've done all of it completely wrong. Can someone help me?
View ArticleExistence of whole number between two real numbers $x$ and $x +1$?
How to prove that there is a whole number, integer, between two real numbers $x$ and $x+1$ (in case $x$ is not whole)?I need this for an exercise solution in my topology class, so I can probably use...
View ArticleDefinition of multiplication of real numbers from product of positive...
--let $A, B \in \mathbb{R}$, with $0\leq A$ and $0 \leq B$, $A \star B :=\{q\in\Bbb Q\mid q<0\}\cup\{a\cdot b\mid a\in A\wedge b\in B\wedge a\ge 0\wedge b\ge 0\}$"this definition is correct:--let $...
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Existence of a continuous curve of zeros
Consider a continuous function $f : [0,1]^2 \to \mathbb R$ with $f(0,y) = 1$ and $f(1,y) = -1$. I want to show that there exists some continuous curve $$(\gamma_1, \gamma_2) = \gamma : [0,1] \to...
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