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.
Best Answer
If you have a sequence of functionals $f_n:X\to\mathbb R$ which is Cauchy in the operator norm, then in particular this implies that for any $x\in X$ the sequence $f_n(x)$ is Cauchy in $\mathbb R$, which means that it has a limit, which we denote by $f(x)$. This gives us a function $f:X\to\mathbb R$ such that $f_n$ converges to $f$ pointwise. Intuitively one might expect that $f$ is then in fact the limit of $f_n$ in $X^*$ - at the very least $f$ is a candidate to be one, and we just need to show that it works.
From there it's actually easy to get a complete proof - it's pretty clear that $f$ is a linear functional, so we just would need to check it's bounded and $f_n$ in fact converge to $f$, which is just a matter of writing down an estimate uniform in $X$.