Note that

- then .
- The quasi-norms are
*rearrangement invariant*.

PropositionIf , then write whereThen

In partiular, whenever .

*Proof:* It suffices to consider of the form with disjoint sets (cf. )

Thus

and so

Things to note: Make continuum problem discrete, concavity of fractional powers

PropositionFor

- The right hand side of part (1) defines an actual norm.
- as Banach spaces, provided

Why not ? *Proof:* For assertion (1)

and a miracle occurs.

For assertion (3), it suffices to consider , which disjoint, and of “norm” 1. Choosing

then

and

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