Let X, Y be Banach spaces and $ X \hookrightarrow Y$ (which would imply $Y' \hookrightarrow X'$). Let the bilinear form $a_0: X \times X \to \mathbb R$ be continuous, symmetric and coercive. Moreover, let the bilinear form $a_1: X \times Y \to \mathbb R$ satisfy $a_1(u,v) \leq C ||u||_X ||v||_Y $ $\forall u \in X, \forall v \in V$
I want to show that the linear mapping $K: X \to X$, $w \to Kw$ defined by $a_0(Kw, v) = a_1(w,v) $ $\forall v \in X$ is compact.
To do so, I divided the claim into 3-4 steps. In the first step, I wanted to show that for $B_1 \in \mathcal L(Y,Z)$ and $B_2 \in \mathcal L(Z,X)$ the operators $B_1 \circ T \in \mathcal L (X,Z)$ and $T \circ B_2 \in \mathcal L(Z,Y)$ are compact, where $X,Y,Z$ are Banach spaces and $T \in \mathcal L(X,Y)$ is a compact operator.
Is this a good idea to start with this step? And how can I show that this really holds true?
Edit: Afterwards, I would define, for $w \in X$, the operator $A_w$ by: $A_w : X \to \mathbb R$, $v \to a_1(w,v)$ and try to show that $A: X \to X'$, $w \to A_w$ is well defined and compact.
But my questions hold: How can I correctly show my first (and second) step?