l-p vector space

[[concept]]

Definition

Matrix Analysis

ℓ^{p}

norm

For $K^{n}$ over $K$ the $ℓ^{p}$ norm for $p \in Z$ is defined as

| | x | |_{p} = {(\sum_{i = 1}^{n} | x_{i} |^{p})}^{1 / p}

When $p = 1$ this is the "manhattan norm"
When $p = 2$ this is the euclidean norm
$| | x | |_{\infty} = max (x_{i})$

Functional Analysis

ℓ^{p}

space

$ℓ^{p} = {a = {a_{j}}_{j = 1}^{\infty} | | | a | |_{p} < \infty} $ $ i s t h e s p a c e o f (i n f i n i t e) s e q u e n c e s . W e d e f i n e t h e $ ℓ^{p} $ n o r m a s $ $ | | a | |_{p} := {\begin{cases} (\sum_{i = 1}^{\infty} | a_{i} |^{p})^{1 / p} 1 \leq p < \infty \\ sup_{1 \leq j < \infty} | a_{j} | p = \infty \end{cases}$

Example

${\frac{1}{j}}_{j = 1}^{\infty} \in ℓ^{p} \forall p > 1$ but not for $p = 1$

References

const modules = await cJS()

const COLUMNS = [  
	{ id: "Name", value: page => page.$link },  
	{ id: "Last Modified", value: page => modules.dateTime.getLastMod(page) },
];  
  
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	let pages = dc.useQuery(queryString);
	
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	return <dc.Table columns={COLUMNS} rows={pages} paging={20}/>;  
}

const { dateTime } = await cJS()

return function View() {
	const file = dc.useCurrentFile();
	return <p class="dv-modified">Created {dateTime.getCreated(file)}     ֍     Last Modified {dateTime.getLastMod(file)}</p>
}