Rearrangement of series. The theorem of Levy-Steiniz. - José Bonet ...

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The example of Marcinkiewicz. Consider the following functions in L2[0, 1]. Here χA is the characteristic function of A. xi,k := χ k k+1 [ 2i , 2i ] , yi,k := −xi,k, 0 ≤ i < ∞, 0 ≤ k < 2 i . Clearly ||xi,k|| 2 = 2 −i for each i, k. One has (x0,0 + y0,0) + (x1,0 + y1,0) + (x1,1 + y1,1) + (x2,0 + y2,0) + ... = 0 x0,0 + (x1,0 + x1,1 + y0,0) + (x2,0 + x2,1 + y1,0) + (x2,2 + x2,3 + y1,1) + ... = 1 No rearrangement converges to the constant function 1/2, since all the partial sums are functions with entire values. José Bonet Rearrangement of series. The theorem of Levy-Steiniz.
The example of Marcinkiewicz. 1 0 1 x00 1 0 1/2 1 x10 1 0 1/2 1 José Bonet Rearrangement of series. The theorem of Levy-Steiniz. x11
Page 1 and 2: Rearrangement of series. The theore
Page 3 and 4: Series � ak. Sequence of real num
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Page 7 and 8: Riemann (1826-1866). Riemann integr
Page 9 and 10: The Theorem of Riemann Idea of the
Page 11 and 12: The alternate harmonic series. An e
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Page 15 and 16: The Theorem Levy Steinitz. Notation
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Page 23 and 24: Series in infinite dimensional Bana
Page 25: The Scottish Cafe, Lvov. 2010. Jos
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Page 39 and 40: Other open problems. Does every non

Rearrangement of series. The theorem of Levy-Steiniz. - José Bonet ...

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