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Halmos in his book (A Hilbert space problem book) says,

1- linear basis, and orthogonal basis of a Hilbert space $H$ have the same cardinality. enter image description here

2- Also he proves if orthogonal dimension of Hilbert space is $N_0$ ( aleph-null ), then its linear dimension is $2^{N_0}$.

My question: In this case orthogonal dimension, and linear dimension are not the same by 2, Is not it a contradiction with 1?

Also if it's possible, give me more information about it.

Please regard me. Thanks in advance.

niki
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    Halmos probably meant "all linear basis have the same cardinality and all orthogonal basis have the same cardinality", which is true. Of course, the other interpretation is false. –  Jun 20 '15 at 11:21
  • @G.Sassatelli : No, I checked it again. Put it's picture above. – niki Jun 20 '15 at 11:26
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    Halmos is referring to bases of the same type there. a) All Hamel bases of a vector space have the same cardinality, and b) all orthonormal bases of a Hilbert space have the same cardinality. He does not intend to say that the linear dimension and the orthogonal dimension are the same. (They are of course the same in the finite-dimensional case, and they are equal in a lot of, but not all infinite-dimensional cases.) – Daniel Fischer Jun 20 '15 at 11:46
  • @DanielFischer: Thanks. – niki Jun 20 '15 at 11:52

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