# Questions on notations about supremum

## 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

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

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Fredrik
Staff Emeritus
$$\sup_{\|x\|=1}\|Tx\|,$$ instead of
$$\sup\big\{\|Tx\|:x\in\mathcal H,\ \|x\|=1\big\}$$ and
$$\sup_{x\in\mathcal H}\frac{\|Tx\|}{\|x\|}$$ instead of
$$\sup\bigg\{\frac{\|Tx\|}{\|x\|}\bigg|\ x \in\mathcal H,\ x\neq 0\bigg\}.$$