Theorem 5.2 (method of continuity) of Gilbarg and Trudinger's "Elliptic partial differential equations of second order" states for $\mathfrak{B}$ a Banach space and $\mathfrak{C}$ a normed linear space, let $L_0$ and $L_1$ be bounded linear operators from $\mathfrak{B}$ into $\mathfrak{C}$. For each $t \in [0,1]$, set $L_t = (1-t)L_0 + tL_1$, and suppose there is a constant $C$ such that \begin{equation} \|x\|_\mathfrak{B} \le C \|L_t x\|_\mathfrak{C} \end{equation} for $t \in [0,1]$. Then $L_1$ maps $\mathfrak{B}$ onto $\mathfrak{C}$ if and only if $L_0$ maps $\mathfrak{B}$ onto $\mathfrak{C}$.
I would like to apply it to show existence and uniqueness of the problems
1) In Theorem 6.8 of the same book, $L_t u := tLu + (1-t)\Delta u, \quad 0 \le t \le 1$, where \begin{equation} Lu = a^{ij}\partial_{x_i}\partial_{x_j}u + b^i\partial_{x_i}u + cu \end{equation} on an open bounded $C^{2,\alpha}$ domain in $\mathbb{R}^n$. Assume $L$ is strictly elliptic, and the coefficients of $L$ are in $C^\alpha(\overline\Omega)$ with $c \le 0$. The authors said "the operator $L_t$ may be considered a bounded linear operator from a Banach space $\mathfrak{B}_1 := \{u \in C^{2,\alpha}(\overline\Omega) \:|\: u = 0 \text{ on } \partial\Omega \}$ into the Banach space $\mathfrak{B}_2 := C^\alpha(\overline\Omega)$". Why is this operator bounded?
2) In Theorem 6.31 of the same book, we want $\mathfrak{L}_t$ to be one-to-one and onto. Is this equivalent to boundedness?
3) What are some equivalent conditions for showing an operator is bounded? I don't see the use of condition $\|L_t u\|_Y \le C \|u\|_X$ for some Banach spaces $X$ and $Y$.
Thank you.