1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Questions on notations about supremum

  1. Feb 26, 2012 #1
    1. The problem statement, all variables and given/known data

    In usual textbooks, what are the meanings of the notations

    (1) [itex]\sup_{k\in\mathbb{N}}|x_{k}|[/itex]

    (2) [itex]\sup_{x\in [ a , b ] }|f(x)-g(x)|[/itex]

    where [itex](x_{k})[/itex] is a real sequence and [itex]f[/itex] and [itex]g[/itex] are real valued functions

    2. Relevant equations


    3. The attempt at a solution

    The meaning I guessed is that

    (1) [itex]\sup_{k\in\mathbb{N}}|x_{k}|=\sup\{|x_{k}|:k\in \mathbb{N}\}[/itex]

    (2) [itex]\sup_{x\in[a,b]}|f(x)-g(x)|=\sup\{|f(x)-g(x)|:x\in[a,b]\}[/itex].
  2. jcsd
  3. Feb 26, 2012 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Your guess is correct.

    I should also mention that when a supremum is written in the shorter form, it is (I believe) common to leave out some information. For example, if T is a bounded linear operator on a Hilbert space ##\mathcal H##, I would write
    $$\sup_{\|x\|=1}\|Tx\|,$$ instead of
    $$\sup\big\{\|Tx\|:x\in\mathcal H,\ \|x\|=1\big\}$$ and
    $$\sup_{x\in\mathcal H}\frac{\|Tx\|}{\|x\|}$$ instead of
    $$\sup\bigg\{\frac{\|Tx\|}{\|x\|}\bigg|\ x \in\mathcal H,\ x\neq 0\bigg\}.$$
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook