For questions on infima.

learn more… | top users | synonyms

0
votes
1answer
13 views

How to prove that the spectral radius of a linear operator is the infimum over all subordinate norms of the corresponding norm of the operator.

I am trying to understand a proof I have seen of the following theorem: $$\rho(A)=\inf_{\|\cdot\|}\|A\|.$$ I understand that to do this, the idea is to show that 1) $\rho(A)\leq\|A\|$ for any norm, ...
0
votes
1answer
74 views

$\exists f:\mathbb{R}\rightarrow \mathbb{R},$ continuous, non-constant, with uncountably many extrema?

I couldnt think of any; by intuition I don't think any can exist, but I can't figure out how to prove it. If it existed then the set of extrema would have to be uncountable but I think this might ...
0
votes
3answers
116 views

Prove that infimum (A)=0 and that supremum (A)=1 in the following set

$$A=\{\frac{n}{m}:m,n \in \mathbb{Z}^+, m>n\}$$ Now, I know that, as $n$ approaches $0$ from above and as $m$ approaches infinity, $\frac{n}{m}$ gets arbitrarily close to $0$, but my professor ...
0
votes
2answers
55 views

Determine whether the following subsets of $\mathbb{R}$ are bounded.

$A=\{x+\frac{1}{x}:x \in (0,\infty)\}$ $B=\{x^2+xy^2:-2 \leq x \leq 1, -1 \leq y \leq 1\}$ I understand the what it means for a set to be bounded above and below, but how would I go about proving ...
3
votes
2answers
161 views

Show that: $inf(A+B)=inf(A)+inf(B)$

Let $A,B$ not empty,bounded subsets of $R$ and $A+B=\{a+b:a \in A,b \in B\}$.Show that: $$\inf(A+B)=\inf(A)+\inf(B)$$ That's what I have tried: -Suppose that $x \in A+B \Rightarrow x=a+b,a \in A,b \in ...
1
vote
1answer
43 views

Supremum calculation

Calculate $\sup(\sum_{k=n+1}^{\infty}\frac{|x_{k}|^{2}}{4^{k} })$, where $x=(x_{1},x_{2},....)$ is a member of $l_{2}$ and the supremum is take over all $x$ with $||x||= 1$. My intuition says the ...
1
vote
0answers
16 views

an example of supremum using relation other than <. like using divides etc??

i want to know some examples of supremum or infimum on a poset using relation, other than $\le$, for example using divides or mod. basiclly i want to know how to find out the supremum of a poset for ...
0
votes
1answer
19 views

Infimum of pre-image of continuous function

Let $f$ be a continuous function on $\mathbb{R}_+$ into $\mathbb{R}$. Then for every $n\in\mathbb{N}$ $$\inf\{t\in\mathbb{R}_+\colon f(t)\in [n,\infty)\}=\inf\{t\in\mathbb{R}_+\colon f(t)\in ...
1
vote
1answer
54 views

Inequality with infimum and supremum for $A \subseteq \bigcup_{n=1}^{\infty}A_n$

Is it true that if $$A \subseteq \bigcup_{n=1}^{\infty}A_n$$ then $$ \sup A - \inf A \le \sum_{n=1}^{\infty} \sup A_n - \sum_{n=1}^{\infty} \inf A_n \quad (\star)$$ I suppose that it is true (I need ...
2
votes
1answer
82 views

Condition For A Set Having A Smallest Element

I am reading the second part of the prolouge of Spivak's Calculus. In the text, he proves the Well-Ordering Principle. Here is a sentence from the book: Suppose that the set A has no least member. ...
2
votes
1answer
144 views

$\inf$ and $\sup$ of a set.

Let $n\geq3$ be an arbitrarily fixed integer. Take all the possible finite sequences $(a_{1},...,a_{n})$ of positive numbers. Find the supremum and the infimum of the set of numbers ...
2
votes
1answer
20 views

Reasoning about Schnirelmann Density: Proving that $d(C) \ge d(A) + d(B)$

I am taking this argument from Gelfond & Linnik's Elementary Methods in the Analytic Theory of Numbers. They state if for every $n \ge 1$, there exists $m \in [1,n]$ where $C(n) - C(n-m) \ge ...
1
vote
1answer
90 views

Bounded functions and infimum/supremum

Question from my homework: ![question] My proof: Let $ S = \{f(x) | x \in \mathbb{R} \} $, since $f$ is bounded it has an infimum & supremum (is this true? I tried to prove it, but have no ...
1
vote
1answer
17 views

Question about Schnirelmann Density and Sumset: if $d(A) \ge \frac{1}{2}$ and $d(B) > 0$, wouldn't $d(A+B)=1$

I've been thinking about the Schnirelmann Density and I think that I may still be confused about SumSet and Density. It seems to me that if $d(A) \ge \frac{1}{2}$ and $d(B) > 0$, then $d(A+{B}) = ...
1
vote
2answers
142 views

Find the supremum, infimum, maximum and minimum

Find the supremum, infimum, maximum and minimum of this set: $$E = \{\frac{2^p}{5^q}:{p \over q} \in (1,2)\text{ and } q > 0\} $$ My thoughts: there is no supremum because we can choose always ...
1
vote
1answer
85 views

Prove that the infimum is not attained for a set $M$

Consider $C([0,1])$ with the $\sup$-norm. Let $$N = \bigg\{ f\in C([0,1]) | \int_0^1 f(x)dx = 0\bigg\}$$ be the closed linear subspace of $C([0,1])$ of functions with zero mean. Let $$X = \{ f\in ...
1
vote
1answer
43 views

Additive properties of sequences: trying to understand Schnirelmann density

I have started reading Gelford & Linnik's elementary methods in analytic number theory (1965). They define a sequence $A$ of integers as: $$0, a_1, a_2,a_3,\dots$$ where $$0 < a_1 < a_2 ...
2
votes
1answer
83 views

Best way to explain how the Infimum and Supremum of this function are obtained…?

I have the function $\;f(x)=\dfrac{x^{(1/2)}}{2+x}\;$ and I know that $\inf(f)$ does not exist and $\sup(f)=2$ but I don't know how to formally show this rigorously? Anyone got a formal way of showing ...
1
vote
0answers
295 views

Find the minimum,maximum, infimum and supremum of sets?

If $X$ is the intersection of all the intervals $(1-\frac{1}{n^2},1+\frac{5}{n^3}]$ for $n=1$ to infinity, what is the minimum, maximum, supremum and infimum of $X$? If $Y$ is the intersection of all ...
2
votes
1answer
58 views

Question on the definition of infimum?

Thm: Let $K$ be compact metric space and $f:K\rightarrow \mathbb{R}$ a continuous real-valued function. Then $f$ is bounded on $K$ and attains its infimum. Since $K$ is compact and $f$ continuous ...
3
votes
1answer
193 views

Let $X$ be a compact metric space. If $f:X\rightarrow \mathbb{R}$ is lower semi-continuous, then $f$ is bounded from below and attains its infimum.

Let $X$ be a compact metric space. If $f:X\rightarrow \mathbb{R}$ is lower semi-continuous, then $f$ is bounded from below and attains its infimum. I want to prove this. This is my proof: Since $X$ ...
0
votes
1answer
74 views

Question about intervals and infima/suprema

Let $L(E)$ be the set of lower bounds of a set $E$ and $(S, \le)$ a partially ordered set. For each $s \in S$, let $$ \langle s] := \{x \in S \mid x \le s\} $$ and $$ [s\rangle := \{x \in S \mid ...
0
votes
0answers
62 views

Infimum of a set of a sequence of numbers

Consider the sequence $\{ y_{n} \}$ of real numbers such that $\sup\{\left | y_{n} \right |:n\in \mathbb{N}\}=4$. Find $\inf \left\{\frac{\left | y_{n} \right |}{n}:n\in \mathbb{N} \right\}$ Since ...
0
votes
2answers
76 views

I need to find the supremum of the set $S$

Let $S:=\{x\ge 0,\sum_{n=1}^{\infty} x^{\sqrt{n}}<\infty\}$; I need to find the supremum of the set $S$. Could any one tell me where to start?
1
vote
2answers
77 views

Troubles calculating a set infimum

I have a set like this one: $$A=\left\{\sqrt{x^2+x}-x, x \in \Re\right\}$$ I am trying to calculate the infimum, which seeing a plot drawn with wolfram has been revealed to be zero (for x=0). ...
1
vote
1answer
126 views

Lim inf of sum of functions

When is the lim inf of sum of two real valued functions equal to the sum of their individual lim inf? That is, I am looking for condition on $f$ and $g$ under which $\liminf\limits_{x \rightarrow ...
2
votes
0answers
69 views

How to obtain the infimum of this inequalities?

Let $A$ be the family of functions $f(z)=z+a_2z^2+\cdots$ that are analytic in unit disk $D:\{z:|z|<1\}$ and $S$ is the subfamily of functions that are univalent in $D$. $R(a)$ is the subfamily of ...
1
vote
0answers
61 views

closest point property of subset of Hilbert space - what are the conditions for existence of inf?

I'm proving the closest point property of a subset of a Hilbert space, ie: $$H$$ is a Hilbert space with a norm generated by the inner product and so on. $$h\in H$$ is a point in H $$M\subset H$$ M ...
6
votes
1answer
1k views

$\inf A = -\sup(-A)$

Let $A$ be a nonempty subset of real numbers which is bounded below. Let $-A$ be the set of of all numbers $-x$, where $x$ is in $A$. Prove that $\inf A = -\sup(-A)$ So far this is what i have ...
3
votes
1answer
71 views

Stability under supremum of sets of social choice function with single peaked preferences

Here is a question emerging from reading Moulin, H. (1980). On strategy-proofness and single peakedness. Public Choice, 35(4), 437–455. The setting is as follows: A non-empty finite set of ...
4
votes
1answer
114 views

Finding the supremum of the following set [duplicate]

I am stuck on the following problem: Let $P=\{x \in \Bbb R: x\ge 0,\sum_{n=1}^{\infty}x^{\sqrt n}< \infty\}$.Then what is the supremum of $P$? Can someone help me out by providing some ...
3
votes
1answer
63 views

Compute infimum

I want to compute the following infimum: $$ \inf\limits_{x_1,\ldots,x_n \geqslant 0} \dfrac{x_1 y_1 + \ldots + x_n y_n}{(a_1 x_1^\alpha + \ldots + a_n x_n^\alpha)^{\frac 1 \alpha}} $$ where $y = ...
1
vote
2answers
57 views

An example of a function which satisfies $\sup_{y}\inf_{x}f(x,y)<\inf_{x}\sup_{y}f(x,y)$

Let $f:X\times Y\to \mathbb{R}$. I could not come up with an example that satisfies $$\sup_{y\in Y}\inf_{x\in X}f(x,y)<\inf_{x\in X}\sup_{y\in Y}f(x,y).$$ Any help would be appreciated. Thanks!
1
vote
1answer
103 views

Question about the infimum of $|f(x)-x|$ , where $f(x)=x$, $x$ is a fixed point of the nonlinear equation.

I am trying to check if the following property holds for fixed points: Suppose: $ f(x)= x $ is given, with solution $x = \theta \gt 0 $ I would like to show : $ \forall \epsilon \in (0,1), ...
0
votes
1answer
45 views

infimum's basic properties in optimization problem

This problem is in Optimizing over some variables slide of Convex Optimization problem. I have a question about basic assumption in this textbook. $$ \inf_{x,y} f(x,y) = \inf_{x} g(x), where, ...
2
votes
0answers
93 views

finding infimum of maximum of two functions

Let $r\geq 1$. Let $C>0$ be a constant. For $x\in R, x>0$ Find the following infimum: $$ \inf_{x}\max\left\{\frac{(1+x)^r}{x}; C\frac{1+x}{x}\right\}. $$
5
votes
1answer
276 views

Show $\sup{A}-\inf{B}=\sup\{a-b:a\in A, b\in B\}$

Let $A, B \subset\mathbb{R}$ be bounded sets. Show $$\sup{A}-\inf{B}=\sup\{a-b:a\in A, b\in B\}$$
1
vote
1answer
191 views

Infimum and limit

I was having trouble with the following question. Any help would be highly appreciated. Let $A$ be the set of K-dimensional vectors with non-negative components. Let $B$ be the set of K-dimensional ...
1
vote
1answer
633 views

Inequalities related to infimum and supremum

Let $f,g: A \rightarrow \mathbb{R}$ be integrable functions on a closed rectangle $A \subset \mathbb{R}^n$. Let $P$ be a partition of $A$ and $S \in P$ a sub-rectangle. Show that: $m_S(f+g) \geq ...
2
votes
1answer
33 views

$f(x) = \inf_{y \in Y} c(x,y) - \inf_{\xi \in X} c(\xi,y) - f(\xi) \Rightarrow f$ is upper semicontinuous

Let $X, Y$ be metric spaces. Given $c: X \times Y \mapsto \mathbb{R}$ continuous, define $$ f(x) = \inf_{y \in Y} \left( c(x,y) - \inf_{\xi \in X} (c(\xi,y) - f(\xi)) \right).$$ Then is $f$ upper ...
2
votes
1answer
301 views

Proof that the infimum of a given set $E$ is unique

Prove that a lower bound of a set might not be unique but the infimum of a given set is unique. Attempt: Consider some $E \subset \mathbb{R} $ such that $E \neq \emptyset$. $E$ is bounded below ...
1
vote
3answers
450 views

Supremum/Infimum proof

Assume that $\inf(A)>0$ and let $A'=\left\{\frac{1}{x} : x\in A\right\}$. I need to show that $\sup(A') = \dfrac{1}{\inf(A)}$. I think this is quite simple, $\sup(A')$ must be $\dfrac{1}{\inf(A)}$ ...
1
vote
1answer
108 views

Infimum of a Hilbert space inner product

This is exercise 5.11 in Brezis's Functional Analysis, Sobolev Spaces, and PDEs. Let $H$ be a Hilbert space, and let $M \subset H$ be a nonzero closed linear subspace. Let $f \in H$, $f \notin ...
2
votes
1answer
43 views

Trying to prove $\text{Hdim}(\bigcup X_i)=\sup_i \text{Hdim}(X_i)$

Suppose $X=\bigcup_i X_i$ is a countable union. I'm trying to prove a statement which wikipedia says follows directly from the definition of Hausdorff Dimension: ...
1
vote
2answers
236 views

Infimum/Supremum of an intersection of subsets of a vector space.

Consider a collection of subsets $A_i$ of an ordnered vector space or field. I am trying to find out, under what (minimal) conditions the following holds. $\inf \bigcap_{i\in I}A_i \le \sup_{i\in ...
3
votes
2answers
92 views

An inequality about sequences in a $\sigma$-algebra

Let $(X,\mathbb X,\mu)$ be a measure space and let $(E_n)$ be a sequence in $\mathbb X$. Show that $$\mu(\lim\inf E_n)\leq\lim\inf\mu(E_n).$$ I am quite sure I need to use the following lemma. ...
11
votes
1answer
160 views

Polynomial $P(x,y)$ with $\inf_{\mathbb{R}^2} P=0$, but without any point where $P=0$

Recently I've came across such problem: give a polynomial $P(x,y)$, with $\inf_{\mathbb{R}^2} P=0$, but there is no point on the plane where $P=0$. I couldn't solve it after a day, and seriously doubt ...
2
votes
1answer
58 views

Continuity of the operation: Infimum of two projections.

The question actually is limited to a very specific case. The following takes place in a fixed Hilbert space. Let $(p_i)_i, (q_i)_i, p, q$ projections (resp. nets of projections) so that ...
1
vote
0answers
227 views

Is the infimum of a continuous function reached?

I am stuck on this problem for a while now, any help would be appreciated. I am working on the proof of a Network Calculus theorem, and I would like to show that the infimum is reached in the ...
2
votes
3answers
324 views

archimedean Property - proof

i am stumbling across this statement. i need to show minimum, maximum, infimum and supremum, if they exist. $$ C:= \bigcup_{n \in \mathbb{N}} [0,1/n[$$ the archimedean property says: let $e,x$ be ...