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While studying the books: Monotone operators in Banach space and nonlinear partial differential equations from Showalter and Quelques méthodes de résolution des problèmes aux limites non linéaires from Lions, in the part of nonlinear evolution equations (for example), they seem to assume that every separable reflexive Banach space has a Schauder basis.

In fact, in chapter 2 of Lions book, when dealing with the equation $$u'(t)+A(u(t))=0, u(0)=u_0,$$

for $u\in L^p((0,T),V)$, where $V$ is a separable reflexive Banach space, Lions use the Galerkin method and assume that there is a Schauder basis for $V$.

The same happen in Showalter's book, for example in Theorem 2.1 in the chapter 2 or in Theorem 4.1 in the chapter 3.

So my question is:

Does every separable reflexive Banach space has a Schauder basis?

It may be the case that I am understanding it wrong, so if this is the case, please let me know.

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In both books, it is not constructed a Schauder basis. In fact, they only assume the existence of a sequence $u_n\in V$ (linearly independent) such that $$\overline{\bigcup_{n=1}^\infty\operatorname{Span}(\{u_1,\cdots, u_n\})}=V. \tag{1}$$

This sequence can be constructed, for example, by using the separability of $V$: let $v_n\in V$ be a dense sequence in $V$ and $V_n=\operatorname{Span}(\{v_1,\cdots,v_n\})$. Take $u_1,\cdots,u_{m_n}$ a basis of $V_n$ and note that the sequence $u_n$ is linearly independent and satisfies $(1)$.

Remark: Not every serapable reflexive Banach space has a Schauder basis. There is Enflo's example.

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  • $\begingroup$ Thank you for sharing this. It may be useful $\endgroup$ – Giuseppe Negro Apr 1 '16 at 14:05
  • $\begingroup$ You are welcome @GiuseppeNegro. I was really confused by it and I thought that it would be good to share this information. I always thought that the sequence used in the Galerkin method was a Schauder basis, but now it is clear that there is nothing to with such a basis. However, by taking a look in the literature, when one has a Schauder basis, or other basis which are more refined, then it is possible to study some convergence properties of the aproximated solutions. $\endgroup$ – Tomás Apr 1 '16 at 14:11
  • $\begingroup$ @GiuseppeNegro, I knew Enflo's example, but in my mind, it was not reflexive. Anyway in Triebel book of functions it is proved that most of the classical functions space has a Schauder basis. $\endgroup$ – Tomás Apr 1 '16 at 14:17
  • $\begingroup$ In fact, for $V$ separable we can construct a linearly independent sequence $(u_n)$ in $V$ such that $\overline{\bigcup_{n=1}^\infty(\{u_1\})}=V$, which is better than $(1)$. Proof here. $\endgroup$ – Pedro Oct 3 '16 at 14:45

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