Questions on notations about supremum

[julypraise]

Homework Statement

In usual textbooks, what are the meanings of the notations

(1) $\sup_{k\in\mathbb{N}}|x_{k}|$

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

where $(x_{k})$ is a real sequence and $f$ and $g$ are real valued functions

Homework Equations

None.

The Attempt at a Solution

The meaning I guessed is that

(1) $\sup_{k\in\mathbb{N}}|x_{k}|=\sup\{|x_{k}|:k\in \mathbb{N}\}$

(2) $\sup_{x\in[a,b]}|f(x)-g(x)|=\sup\{|f(x)-g(x)|:x\in[a,b]\}$.

 

[Fredrik]

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