[In what follows = 1 by convention.]
Is there some closed infinite dimensional linear subspace of
such that belongs to
for all in ?
This problem is related to the Erdos - Shapiro - Shields paper [ESS].
From this paper it follows that the answer is negative if is replaced by .
Some thoughts. Suppose that such an exists, and take some .
Let be in .
Then clearly is in , too, hence
belongs to .
Next, it is easy to see that .
Therefore, is contained in as a linear subspace
(i.e., algebraically).
Now, applying the Closed Graph Theorem to the natural linear embedding
, it follows that is
continuous. Consequently, the Hilbertian 2-norm and the -norm
are equivalent on . Moreover, it follows that is complete
w.r.t. the -norm, and, in turn, it is a closed subspace of .
And this is true for all .
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