Let $\mathcal{B}$ be a Banach space, and $V$ a normed linear space. $L_0,L_1:\mathcal{B}\to V$ are bounded linear operators. Assume $\exists c$ such that $L_t := (1-t)L_0 + tL_1$ satisfies:
$$(*)\quad \|x\|_\mathcal{B} \leq c\|L_t x\|_V,\quad \forall t\in [0,1].$$
Proposition: $L_0$ is onto $\iff$ $L_1$ is.
I want to prove this. So I can see if we assume $L_s$ is surjective for some $s\in[0,1]$ we get that $L_s$ is bijective from linearity and $(*)$ and we get an inverse operator from $(*)$ and surjectivity, $L_s^{-1}$.
I don't know how to proceed though. Can I have some guidance please.
The point of the standard proof to this (which can be found, e.g., in Gilbarg Trudingers 'Elliptic Partial differential equations of Second order', Theorm 5.2) is to assume that $L_s$ is onto for some $s$ and then show that this implies $L_t$ is for all $t$ in a controlled neighbourhood of $s$. By covering $[0,1]$ with finitely many such intervals you can show that the set where $L_s$ is onto is connected.
To show the statement I indicated above you can reformulate the equation $L_t x = y$ as a fixed point problem for $T$ given by $$Tx := L_s^{-1}y + (t-s) L_s^{-1}(L_0 - L_1) x $$
Now for that $T$ you can verify the prerequisites of the Banach fixed point theorem if $t$ is close enough to $s$.
If this is not enough hint for you you can always look it up (see the source mentioned above)
(Edit: as indicated, the point is always to show that the set where some property for some continuos family of operations you want to be true is connected. This usually means you want to show it's open and closed. Often, open is more or less easy, since, for example, the set of invertible maps is often open in reasonable setups. To show that the set is closed, you usually try to apply some compactness result. This is where, e.g., in elliptic PDE the compact embeddings of Sobolev spaces or the Schauder estimates kick in. In the proof I referred to applying compactness is somewhat hidden behind the clever observation that the applicability of the Banach fixed point theorem can be shown to be true on intervals of controlled size (controlled by the norms of $L_0$ and $L_1$, to be more precise)).