Questions tagged [supremum-and-infimum]

For questions on suprema and infima. Use together with a subject area tag, such as (real-analysis) or (order-theory).

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Finding the supremum, infimum, and bounds of $f(x) = x^2$ for $x \le a$ and $f(x) = a + 2$ for $x > a$ in the interval $(-a-1,a+1)$, where $a > -1$.

My current progress; $x^2$ is increasing, so it will always have a minimum of $0$. Furthermore, if $a > -1$, then $a + 2 > 1$ and $-a -1 < 0$. Therefore, $f(x) \ge 0$. I tried to split $f$ ...
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Set of measures on the simplex of matrices that take support on rank 1 matrices has a minimal element

Let $M=\{ A\in\mathbb R^{n\times m} : \forall i,j~A_{i,j}\geq 0,\sum_{i,j} A_{i,j} = 1 \}$ be the simplex of $n\times m$ matrices, it was shown in a previous question that the set $R$ of rank $1$ ...
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Prove that the set of reciprocals of natural numbers has no positive infimum.

Prove that for each $x>0$, there is an $n \in \mathbb N$ such that $\frac{1}{n}$ $<$ $x$, WITHOUT using the Archimedean theorem. Its simple enough with the theorem, but without the theorem I ...
• 165
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ways to calculate norm of the Bounded Linear Functional.

$\Lambda$ be a linear functional on $C([0,1])$ defined by $$\Lambda(f) = \int_0^1 xf(x)dx \;\;\; \text{ for } f \in C([0,1]).$$ and use $\|f\|_{sup} = \sup_{x \in [0,1]} |f(x)|$ for $f \in C([0,1])$...
• 521
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Supremum of bounded family of functions

Let $X$ be a metric space and $\mathcal{F}$ family of functions from $X$ to $\mathbb{R}$. If $\mathcal{F}$ is bounded above in $X$, then $\sup_{f\in \mathcal{F}} f(x)<\infty$. So far, I read in ...
• 263
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Supremum with respect to an object

In my Real Analysis class I've often seen the Upper Riemann Sum (in the definition of Riemann integrability) defined as: $$\sup_P\{U(f,P)\}$$ This seems to mean the supremum of all possible upper sums ...
• 101
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Let $A \subset \mathbb{R}$ and $c \in \mathbb{R}$. If $c \geq 0$, then $\sup{cA} = c\sup{A}$. How to prove this theorem? I am thinking of deriving to the inequalities $\sup{cA} \geq c\sup{A}$ and $\... • 121 0 votes 0 answers 59 views supremum of integral over probability measures Let$f,g,h: (\mathbb{S}^n)^3 \to \mathbb{R}$be some bounded and continuous functions such that:$f(\theta,\varphi,\psi)=g(\theta,\varphi,\psi)+h(\theta,\varphi,\psi)$for all$(\theta,\varphi,\psi)\...
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(The second part of the solution is confusing me) The Question: A is a bounded subset of ℝ B = {b|b = 2a + 3, a ∈ A} Prove that Sup(B) = 2Sup(A) + 3 Part 1 of solution: We get 2a + 3 ≤ 2Sup(A) + 3 So ...
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Almost sure convergence for absolute value of random variables and inequalities and supremums

I read this article about almost sure inequality and the supremum from Almost sure inequality and the supremum Let $X_n$ be a sequence of random variables, if $\left|X_n \right| \leq Y$ almost surely ...
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If $X$ is a compact space and $Y$ a metric topological space with metric $d$, then $d'(\alpha,\beta) = sup_{x \in X} d(\alpha (x),\beta (x))$ is a metric on $TOP(X,Y)$ $TOP(X,Y)$ denotes the set of ...
The exact problem goes like: Let some $A$ be a bounded and non-empty set. There is a sequence ${x}$ which converges to some point $x'$ in $A$ and I have to prove that $x'$ is the max($A$) given that ...