Suppose we have a Banach space $X$ and its dual $X^{'}$. Then the duality pairing is often written $$\langle f,v\rangle_{X^{'},X} = f(v)$$
for $f \in X^{'}$ and $v \in X$. But is it allowed to write
$$ \langle v,f\rangle_{X,X^{'}} $$ to mean $\langle f,v\rangle_{X^{'},X}$?
In the setting of a Hilbert space $H$, for a given linear operator $A:H\to H^{'}$, I saw $A$ being self-adjont is written as
$$ \langle Av,u\rangle = \langle v,Au\rangle $$
Does $\langle v,Au\rangle$ here actually mean $\langle Au,v\rangle= Au(v)$?
