# 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).

2,012 questions
Filter by
Sorted by
Tagged with
31 views

49 views

28 views

### Can the infimum of a strictly positive functional from $l_\infty$ on the nonegative part of the unit sphere in $l_1$ be equal to zero

**Let $p=(p_1,p_2,...)\in l_\infty$ satisfying $p_i> 0$ for all $i$ and $S_+=\{x=(x_1,x_2,...)\in l_1, x_i\geq 0, \forall i, \sum_ix_i=1\}$. Denote $<p,x>=\sum_ip_ix_i$. Is it possible ...
30 views

### upper and lower limits of a number sequence

I've got a number sequence, $a_n = \frac{1+(-1)^n 2n}{1+3n}$ and I have to calculate the upper limit and the lower limit. First of all, I've divided the sequence in two subsequences: the even ...
37 views

### A monotone function $f$ on $[0,1]$ satisfying $f\big(\frac14\big)f\big(\frac34\big)\lt 0$

Let $f:[0,1]\rightarrow \mathbb{R}$ be a monotonic function with $f\big(\frac14\big)f\big(\frac34\big)\lt 0$ .Suppose sup$\{x\in [0,1]: f(x)\lt 0\}=\alpha$ Which of the following statements are ...
76 views

### Are there any properties of sup?

Can I say: $$\sup |f(x)-h(x)+g(x)-g(x)| = \sup|f(x)-g(x)|+ \sup|g(x)-h(x)|$$ I can't seem to find any properties of $\sup$ , so i am wondering if there is any properties that we can generalize to ...
36 views

15 views

### Differentiable approximation of supremum of a function

I am trying to solve an optimization problem of the form: min$_y$ f(y), s.t. g(y,t)$\leq0$ ∀ t∈D. As I can calculate the gradients for f and g, I was thinking of using a penalty method, in which ...
### Show that in general $\inf \sup \ne \sup \inf$ for bilinear functions
I am working on the following exercise: Let $K \subseteq \mathbb{R}^n$, $L \subseteq \mathbb{R}^m$ and let $F(K,L) \rightarrow \mathbb{R}$ with  F(x, \lambda) := c^Tx + \lambda^Tb - \lambda ...