Sunday, 1 July 2007

fa.functional analysis - Subspaces of L2L2

[In what follows 0000= 1 by convention.]



Is there some closed infinite dimensional linear subspace FF of L2(0,1)L2(0,1)
such that left|fright|left|fright|left|fright|left|fright| belongs to L2(0,1)L2(0,1)
for all ff in FF ?



This problem is related to the Erdos - Shapiro - Shields paper [ESS].
From this paper it follows that the answer is negative if left|fright|left|fright|left|fright|left|fright| is replaced by left|fright|left|fright|2left|fright|left|fright|2.



Some thoughts. Suppose that such an FF exists, and take some p>2p>2.
Let ff be in FF.



Then clearly g:=(p/2)cdotfg:=(p/2)cdotf is in FF, too, hence h:=left|gright|left|gright|h:=left|gright|left|gright|
belongs to L2(0,1)L2(0,1).



Next, it is easy to see that leftVertfrightVertppleq1+leftVerthrightVert22<+inftyleftVertfrightVertppleq1+leftVerthrightVert22<+infty.



Therefore, FF is contained in Lp(0,1)Lp(0,1) as a linear subspace
(i.e., algebraically).



Now, applying the Closed Graph Theorem to the natural linear embedding
j:(F,||.||2)rightarrowLp(0,1)j:(F,||.||2)rightarrowLp(0,1), it follows that jj is
continuous. Consequently, the Hilbertian 2-norm and the pp-norm
are equivalent on FF. Moreover, it follows that FF is complete
w.r.t. the pp-norm, and, in turn, it is a closed subspace of Lp(0,1)Lp(0,1).



And this is true for all p>2p>2.



[ESS] http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.mmj/1028999306

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