l-p vector space

[[concept]]

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 $p = 1$

Mentions

File	Last Modified
Functional Analysis Lecture 1	2025-10-02
Functional Analysis Lecture 13	2025-10-02
Functional Analysis Lecture 15	2025-10-02
separable Hilbert spaces are bijective with ell-2	2025-10-02

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>
}