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I am reading the paper http://arxiv.org/abs/quant-ph/0502053 listed in the reference of Wikipedia Rigged Hilbert Space. I have a question about the relation, Φ ⊂ H ⊂ Φ', where H is Hilbert space, Φ is its subspace and Φ' is dual space of Φ.

Φ⊂H and Φ⊂Φ' are obvious. How can we say H ⊂ Φ' ? Φ' has nothing to do with the elements of H that is not in Φ, I assume. Your advice is appreciable.