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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


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    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
    $$\sup\bigg\{\frac{\|Tx\|}{\|x\|}\bigg|\ x \in\mathcal H,\ x\neq 0\bigg\}.$$
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