Linked Questions
18 questions linked to/from $X$ compact metric space, $f:X\rightarrow\mathbb{R}$ continuous attains max/min
83
votes
6
answers
4k
views
If a two variable smooth function has two global minima, will it necessarily have a third critical point?
Assume that $f:\mathbb{R}^2\to\mathbb{R}$ a $C^{\infty}$ function that has exactly two minimum global points. Is it true that $f$ has always another critical point?
A standard visualization trick is ...
- 135k
14
votes
2
answers
8k
views
Any lower semicontinuous function $f: X \to \mathbb{R}$ on a compact set $K \subseteq X$ attains a min on $K$.
I've been thinking about this problem for a long time right now, and feel stuck.
Given that $X$ is a topological space, and that for $f$ to be lower semicontinuous, for any $x \in X$ and $\epsilon &...
- 1,543
18
votes
2
answers
1k
views
A non-compact topological space where every continuous real map attains max and min
Today I learnt in class that if $X$ is compact then any continuous map $f:X\to\mathbb{R}$ attains max and min. I was thinking if the converse is true:
If every continuous map $f:X\to\mathbb{R}$ ...
- 10.7k
-1
votes
3
answers
978
views
Prove that there exists a point $a$ in $A$ such that $| c-a | =\inf \{| c-x |: x \in A \}$?
Let $A$ be a nonempty compact subset of $\mathbb{R}$ and $c \in \mathbb{R}$.
Prove that there exists a point $a$ in $A$ such that $| c-a | =\inf \{| c-x |: x \in A \}$?
- 65
3
votes
2
answers
168
views
Largest $k$ such that $k \operatorname{Tr} H^2 C \le (\operatorname{Tr} H C)^2$
Suppose we are given $H$, a positive semidefinite $d\times d$matrix. How do I find $k$, the largest $k'$ such that the following holds for all positive semidefinite $C$ with unit trace?
$$k' \...
- 5,489
0
votes
5
answers
3k
views
Find absolute maximum and minimum of the function on the given domain.
Given $f(x,y)=2x^4-xy^2+2y^2,0\le x\le 4, 0\le y\le2$. Find absolute extrema of $f(x,y)$.
I have found $\partial f/\partial x=8x^3-y^2, \partial f/\partial y=-2xy+4y$ and after solving the equation ...
- 9
4
votes
2
answers
353
views
Path connected compact set with given property
Problem:
Suppose $K$ is a compact subset of $\Bbb R^n$, and that for all $k_1, k_2 \in K$, there exists a continuous function $p:[0,1] \rightarrow K$ such that $p(0) = k_1 $ and $p_1 = k_2$. Then let $...
- 3,848
4
votes
2
answers
206
views
Proof that if a function has a limit for large x, then the function is bounded.
I need help in order to confirm whether my proof is approved or not. It follows as:
Claim: Let $f: [a,\infty) \mapsto \mathbb{R}$ where $f$ is continous. If $\exists \lim_{x \rightarrow\infty}f(x)=L$ ...
- 1,847
0
votes
1
answer
976
views
If $S$ is compact, show that there exists a point $x\in S$ such that $d(x,T)=d(S,T)$
$S,T\subset X$.
$S$ is compact.
$d(S,T):=inf\{d(s,t): s\in S, t\in T\}$.
My idea is to look at the set $A:=\{d(s,T), s\in S\}$, where by definition $d(s,T):=inf\{d(s,t),t\in T\}$
I know that $S$...
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3
votes
3
answers
151
views
Existence of such points in compact and connected topological space $X$
Let $X$ be a topological space which is compact and connected.
$f$ is a continuous function such that;
$f : X \to \mathbb{C}-\{0\}$.
Explain why there exists two points $x_0$ and $x_1$ in $X$ such ...
- 31
0
votes
1
answer
369
views
Why do we need the notion of a compact set to define a game?
$\textit{Definition:}$ $I$ is a finite set of players and and $G=((S^i)_i,g)$ is a compact game, that is given by a compact set of strategies $S^i$ for each player $i$ and by a continuous payoff ...
- 429
1
vote
2
answers
303
views
Show that if $|f(z)| \leq M$ for $z \in \partial D$ for $z \in \mathbb{C}$ and M being a constant, then $|f(z)| \leq M$ for all $z \in D$
We are given that $f$ is analytic in a bounded domain $D$ and continuous on the boundary $\partial D$. Also we can assume that $|f(z)| \leq M$ for $z \in \partial D$ for $z \in \mathbb{C}$.
I'm unsure ...
- 904
5
votes
5
answers
121
views
Given $x_{0}$, let $f(x) = \|x-x_{0}\|$. Show tha $f$ has a minumum on any closed, nonempty set $A \subset \mathbb{R^{n}}$
Given $x_{0}$, let $f(x) = \|x-x_{0}\|$. Show tha $f$ has a minumum on any closed, nonempty set $A \subset \mathbb{R^{n}}$.
I tried to do a test for reduction to the absurd, but it was a little ...
- 541
2
votes
2
answers
58
views
about Euclidean Space
Suppose $E_1,E_2 \subseteq \Bbb R^m$ are closed sets and at least one of them is a bounded set.
Prove that there exist $x_0\in E_1,y_0\in E_2$,such that $\rho (x_0,y_0)=\rho(E_1,E_2)$
Attempt :I ...
- 453
0
votes
1
answer
152
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Proof that compact sets in a metric space have finite diameter.
Let $(X,d)$ be a metric space and $K$ a compact set in $X$. If $x\in K$, let $\mathcal{O}_x$ be a neighborhood of $x$ with finite diameter. Since $K$ is compact, there are finitely many points $x_1,\...
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