# Galerkin method and Schauder basis.

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.

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 $(1)$.
• 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. – Pedro Oct 3 '16 at 14:45