Biorthogonal functions in $L^p$ I asked one question that is already answered:
1.) I have a question about Lemma 9.5 on page 93/94 reference.
It's about the part of the proof where the sequence of $(g_n^*)$ are introduced. I don't understand how they are defined and I don't understand what this has to do with the fact the sequence $(g_n)$ is a monotone basic sequence?
Could anybody here explain to me how they are defined?  ( This question is already answered due to Umberto )
Now my second question is the following and this is what the bounty is about:
2.) And does anybody know how the result on page 94 after the lemma on the beginning of the second half of the page is concluded from the preceding lemma ( I'm talking about the one that starts with: It's easy to modify...)? I don't see how they (with all details) come up with this result. 
The fact that you can approximate the norm one functions by functions with small support is explained on page 93 (middle part). Further important references are the theorem 4.7 and Lemma 4.3. 
You can find the whole reference here A short course on Banach space theory
If anything is unclear, please let me know.
My lemma is the following:
Let $(f_n)_{n \in \mathbb{N}} \subseteq L^p$ with $p \in [1, \infty)$ and $f_n \notin M(p, \frac{1}{8^n}=: \varepsilon_n)$ where we also have \ $||f_n||_p=1$, then there is a subsequence $(f_{n_k})_{k \in \mathbb{N}},$ that is equivalent to a sequence $(v_k)_{k \in \mathbb{N}} \subseteq L^p$, where all elements have disjoint support. In particular we have that $\overline{\text{span}}\{f_{n_k};k \in \mathbb{N} \}$ in $L^p$ is complemented and isomorphic to $\ell^p$.
Proof: We know that we can write every $f_n$ as $f_n = g_n + h_n$ , where $g_n,h_n \in L^p$, $\mu(supp(g_n))< \varepsilon_n$ and $||h_n||_p< \varepsilon_n$. From the triangle inequality we get $1 - \varepsilon_n < ||g_n|| < 1 + \varepsilon_n $. Let  $A_n:= \text{supp}(g_n)$ for all $n \in \mathbb{N}$, then we are able to $\mu(A_n) \xrightarrow[n \to \infty]{}0$ construct (inductively) a subsequence$(g_{n_k})_{k \in \mathbb{N}}$, such that
for all $k \in \mathbb{N}$ we have
\begin{equation}
\int_{A_{n_{k+1}}} \sum_{i=1}^k |g_{n_i}|^p d \mu < \frac{1}{8^{(k+1)p}}.
\end{equation}
Take $B_k := A_{n_k} \backslash \bigcup_{i=k + 1}^{\infty} A_{n_i}$ and  take $v_k:=g_{n_k} \chi_{B_k}$. Then we have that $A_{n_k} \backslash B_k \subseteq \bigcup_{i=k + 1}^{\infty} A_{n_i}$.  By construction we have that $i \neq j$ then we have $B_i \cap B_j = \emptyset$ and therefore do all functions $v_k\in L^p $ have a disjoint support. Then we have for all $k \in \mathbb{N}$
\begin{equation}
\begin{aligned}
||g_{n_k} - v_k||_p^p  &  = \int_{A_{n_k} \backslash B_{k}} |g_{n_k}|^p d \mu \\
                        & \le \int_{\bigcup_{i=k + 1}^{\infty} A_{n_i}} |g_{n_k}|^p d \mu \\
                        & \le \sum_{i = k + 1}^\infty \int_{A_{n_i}} |g_{n_k}|^p d \mu \\
            & \le \sum_{i = k + 1}^\infty \frac{1}{8^{i p}}  \\
                        & \le \frac{1}{8^{k p}}  \\
\end{aligned}
\end{equation}
and therefore $||g_{n_k} - v_k||_p \le \frac{1}{8^{k}}.$ Hence we have $||v_k||\le ||g_{n_k}|| \le 1+ \varepsilon_{n_k}$ and by using the triangle inequality we get $||g_{n_k}||_p \le ||g_{n_k} - v_k||_p + ||v_k||_p$ and therefore immediately $||v_k||_p \ge 1 - \frac{2}{8^k} \ge \frac{3}{4}$. If we now look at the biorthogonal functions to $(v_k)_{k \in \mathbb{N}} \subseteq L^p$ called $(v'_k)_{k \in \mathbb{N}} \subseteq L^q$, then we have due to the disjoint support of the $v_k$ that for arbitrary $(a_k)_{k \in \mathbb{N}}$ and $n,m \in \mathbb{N}$ with $m \ge n$ 
\begin{equation}
\left| \left| \sum_{k=1}^{m} a_k v_k \right| \right|_p^p= \int_X \left| \sum_{k=1}^{m} a_k v_k \right|^p d \mu = \sum_{k=1}^{m} |a_k|^p \int_X \left| v_k \right|^p d \mu = \sum_{k=1}^{m} |a_k|^p ||v_k||_p^p \ge |a_n|^p |v_n|^p.
\end{equation}
By taking the limit we have that for all $n \in \mathbb{N}$ and $x \in \overline{\text{span}}\{v_k; k \in \mathbb{N} \}$for which we have a unique coefficient sequence $(a_n)_{n \in \mathbb{N}}$, $||x||_p = \left| \left| \sum_{i=1}^{\infty} a_i g_i \right| \right|_p \ge |a_n|||g_n||_p$ gilt. \
 Since we have $L^p' \cong L^q$ one gets
\begin{equation}
||v_k'||_q = \sup_{||x||_p = 1} |v_k'(x)|  \le  \sup_{||x||_p = 1} |a_k| \le  \sup_{||x||_p = 1} \frac{||x||_p}{||v_k||_p} \le  \frac{4}{3}.
\end{equation}
Therefore
\begin{equation}
\begin{aligned}
\sum_{i=1}^{\infty} ||v_i'||_q ||f_{n_i} - v_i||_p &\le \sum_{i=1}^{\infty} ||v_i'||_q \left(||f_{n_i} - g_{n_i}||_p  + ||g_{n_i} - v_i||_p\right) \\
&\le \sum_{i=1}^{\infty} ||v_i'||_q \left( \frac{1}{8^i}  + \frac{1}{8^i} \right) \\
&\le \frac{8}{3} \sum_{i=1}^{\infty} \frac{1}{8^i} \\
&= \frac{8}{3}  \cdot \frac{\frac{1}{8}}{1 - \frac{1}{8}} \\
&= \frac{8}{21} < 1,
\end{aligned}
\end{equation}
and by the theorem of small perturbations, the lemma is true.
\qed
 A: 1) The lemma you refer to introduces the $g_n^*$ and uses them in an application of Theorem 4.7.
2) Theorem 4.7 calls them "coordinate functionals"
3) The index entry for "coordinate functionals" points to page 25.
4) Coordinate functionals are defined on page 25.
Edited to add:
Think of $g_n^*$ as a functional on $L^p$. If $\{g_n\}$ is a basic sequence in $L^p$ and $x = \displaystyle \sum a_n g_n$, then $g_n^*(x) = a_n$. The basic projections $P_n : L^p \to L^p$ are defined by $$P_n(x) = a_1 g_1 + \ldots + a_n g_n.$$ If the $\{g_n\}$ have disjoint support there is a lot more than can be said. In this case at most one $g_n$ is nonzero and $$|x|^p = \sum |a_n|^p |g_n|^p$$ so that each $a_n$ satisfies $|a_n| \le \dfrac{||x||_p}{||g_n||_p}$. 
The norm of the functional $g_n^*$ can be computed using the formula
$$ ||g_n^*||_q = \sup_{||x||_p=1} g_n^*(x).$$ In the proof you are reading each $||g_n||_p \ge \dfrac 34$, so that if $||x||_p = 1$ we have $$|g_n^*(x)| = |a_n| \le \frac{1}{3/4} = \frac 43.$$ Thus each $||g_n^*|| < \dfrac 43$. (The book gives $8/3$ but I think the extra factor of $2$ is unnecessary).
A: The assumption that $\{f_n\}$ is seminormalized means there are constants $A > 0$ and $B$ with the property that $A \le \|f_n\|_p \le B$ for all $n$. 
The short answer to your question is to select the subsequence $\{f_n\}$ so that
$$
\int_{A_{n+1}} \sum_{i=1}^n |f_i(t)|^p \, dt < A^p 4^{-(n+1)p}$$ for all $n$. 
To expand on this, 
if you look at the proof of Lemma 9.5, the construction of the subsequence of $\{f_n\}$ and the definition of $\{g_n\}$ did not use the fact that each $f_n$ has norm $1$. Just modify the proof so that $\|f_n - g_n\|_p^p < A^p 4^{-np}$ instead of  $\|f_n - g_n\|_p^p < 4^{-np}$.
Since each $g_n$ is defined as $f_n \cdot \chi_{B_n}$ you always have $\|g_n\| \le \|f_n\| \le B$. On the other hand, $\|g_n\|_p \ge \|f_n\|_p - \|f_n - g_n\|_p \ge A - A 4^{-n}$. Thus $A - A4^{-n} \le \|g_n\| \le B$ for all $n$.  As in the preceding answer, you can now estimate the norms of the coordinate functionals as   $$\|g_n^*\|_q \le \frac{1}{A - A 4^{-n}}.$$ Consequently  $$\|g_n^*\|_q \|f_n - g_n\|_p \le \frac{A 4^{-n}}{A-A4^{-n}} = \frac{1}{4^n  - 1}.$$
This makes the sum small enough to apply the principle of small perturbations.
