the cartesian product of banach spaces is banach

[[concept]]

Corollary

If $B_{1}, B_{2}$ are Banach spaces, then $B_{1} \times B_{2}$ with norm $∣∣(b_{1}, b_{2})∣∣ := ∣∣ b_{1} ∣∣ + ∣∣ b_{2} ∣∣$ is a Banach space.

Proof

Exercise

just need to check all the definitions for the norm

check that Cauchy sequences in $B_{1} \times B_{2}$ consist of Cauchy sequences in each of $B_{1}$ and $B_{2}$ (similar to completeness of $R^{2}$ )

Mentions

Mentions

TABLE file.mday as "Last Modified"
FROM [[]]
 
FLATTEN choice(contains(artist, this.file.link), 1, "") + choice(contains(author, this.file.link), 1, "") + choice(contains(director, this.file.link), 1, "") + choice(contains(source, this.file.link), 1, "") as direct_source
 
WHERE !direct_source
SORT file.mday ASC
SORT file.name ASC

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>
}