Theorem. Let $X$ be a normed space and $Y$ be a Banach space. Then the set of continuous linear maps $L(X,Y)$ is a Banach space (with the operator norm).
From the well-known theorem above, we get an immediate consequence: Dual spaces $X^*$ are always Banach spaces.
Why should that be true intuitively? (I am not looking for a proof.)
I'd like to think that somehow, the space $L(X,Y)$ plays more on $Y$ than on $X$, so it inherites being Banach from $Y$. Though, I can't quite make out a clear intuition for all this.