Questions tagged [supremum-and-infimum]

Question 1

For my question I am trying to simplify the setting of concern as much as possible. Therefore, let $(\Omega, \mathcal{A}, P)$ be a probability space and $X:(\Omega, \mathcal{A})\rightarrow (\mathcal{X}...

Question 2

In the following example the $M_n = \text{sup} \{|f_n(x)-f(x)| : x \in \mathbb{R}\}$ does not exist for any $ n \in \mathbb{N}$ $$ f_n(x) = \frac{x}{n} \text{ for all} \, x \in \mathbb{R} $$ $f(x) = ...

Question 3

In my lecture notes there is basic examples with no proof for diameter of certain intervals, I tried to prove generally the diameter for any open interval $(a,b)$, in $\mathbb{R}$ with metric $|x-y|$, ...

Question 4

I encountered the following situation. Let $V,W$ be Hilbert spaces and $d$ a bilinear form. I have the following condition: $$\sup_{v \in V} \frac{d(\mu,v)}{\|v\|} \geq \beta \|\mu\| \quad \forall \mu ...

Question 5

For every natural number $d \in \mathbb{N}$, let $$\mathcal{T}(\mathbb{R}^d)=\{[a_1,b_1) \times\dots \times[a_d,b_d): a_i,b_i\in \mathbb{R}\}$$ denote the set of $d$-dimensional boxes in $\mathbb{R}^d$...

Question 6

Find all real numbers $k$ for which the inequality $\frac{1}{x^2 + y^2 + 1} \leq k$ holds for all $x, y$ satisfying $-1 < x < 1$ and $-1 < y < 1$. My initial thought was to find the ...

Question 7

Let $F\subset C[0,1]$ be a linear subspace which contains the constant functions and separates points of $[0,1]$. Assume that for every $f\in C[0,1]$ and every $x\in[0,1]$ we have $$ f(x)=\sup\{g(x)\...

Question 8

Let $f(n)$ be a real-valued sequence defined for $n \in \mathbb{N}$, with $f(n) > 0$ for all $n$. Define a new sequence: $$ g(n) = \log_b(f(n)) $$ I know that when $0 < b < 1$, the ...

Question 9

I have this exercise: Determine the lower and upper bounds of the following numerical set, specifying whether they are a maximum and/or minimum: $$ X = \{\arctan(n^2 - 7n - 1) : n \in \mathbb{N}\} $$ ...

Question 10

A question from some analysis homework I'm doing asks the following (copied exactly): Let $S$ and $T$ be nonempty subsets of $\mathbb{R}$ such that $S \subset T$. Prove $$ \inf T \leq \inf S \leq \sup ...

Question 11

We are given that $n$ is a positive integer and $t \geqslant 1$. Let $u_1, u_2, \ldots, u_n$ be complex numbers and let $a_1, a_2, \ldots, a_n$ be complex numbers such that ${\operatorname{Re}} (a_i) \...

Question 12

(See the motivation for more info.) Suppose $\dim_{\text{H}}(\cdot)$ is the Hausdorff dimension, and $\mathcal{H}^{\dim_{\text{H}}(\cdot)}(\cdot)$ is the Hausdorff measure in its dimension on the ...

Question 13

In my course of Calculus I, we began to study bounds and some properties of the supremum and infimum. In my homework, there is a problem involving supremum and infimum properties under scalar ...

Question 14

Let $\emptyset$ be the empty set. Suppose $\dim_{\text{H}}(\cdot)$ be the Hausdorff dimension, where $\mathcal{H}^{\dim_{\text{H}}(\cdot)}(\cdot)$ is the Hausdorff measure in its dimension on the ...

Question 15

I am currently following an online lecture attempting to understand Borel sets < https://youtu.be/NdZnhTMoNnM?si=L-yrvxP0TIlWJfNi>. I understand the motivation behind taking the infimum when ...

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Questions tagged [supremum-and-infimum]

Measurability of function minimizing an integral

Can we find a sequence of functions for which $\text{sup} \{|f_n(x)-f(x)| : x \in \mathbb{R}\}$ does not exist but is uniformly convergent? [closed]

Prove the diameter of open interval $(a,b)$ is $b-a$.

Restrict supremum to a subset

Proving the equality of the two suprema

Find all values of $k$ for which the inequality holds for all $x, y$ in the interval $(-1, 1)$

When does the supremum over a subspace $F\subset C[0,1]$ commute with integration?

How do transformations by exponential and logarithmic functions affect monotonicity and extrema of a sequence?

Analyzing bounds and extremes of an arctangent sequence defined by a quadratic argument

Does this proof need an extra condition to work?

Bound of a Rayleigh quotient over the span of multiple exponents

Is the following set empty and can its infimum be equivalent to positive infinity?

Proving properties of supremum and infimum under scalar multiplication

What is the Hausdorff measure of the empty set in its dimension?

Understanding the implications of using the infimum when evaluating the outer measure of a set.

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