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Questions on notations about supremum

  • Thread starter julypraise
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  • #1
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Homework Statement



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



Homework Equations



None.



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].
 

Answers and Replies

  • #2
Fredrik
Staff Emeritus
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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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