Recently I found this statement -- the gradient operator is the adjoint of the minus divergence operator -- in one of my lecture notes. Knowing only a little about functional analysis, I'm looking for an intuitive interpretation.

I already found this topic which has a few good answers, but I'd like to view it from a somewhat different angle.

So say there are two vector spaces $X$ and $Y$, and that $L$ is a linear operator that maps elements from $X$ to $Y$, i.e. $L:X\to Y$. Furthermore, the set of all bounded/continuous linear functionals on $X$ (i.e. $f\;| f:X\to\mathbb{R}$) is called the dual (space) of $X$, marked $X'$. Mutatis mutandis for $Y'$.

Then, apparently if we take a look at our $L$, there is an operator (called the adjoint operator) $L'$ that maps elements from $Y'$ to $X'$, right? So $L':Y'\to X'$.

To make this a little less abstract, let's take the gradient operator $\nabla$. Now, I don't know how to write down appropriate spaces $X$ and $Y$ such that $\nabla:X\to Y$. I thought about $C^1[a,b]$, the space of continuous differentiable functions on the interval $[a,b]$, which is then mapped to $C^0[a,b]$. But these are only functions of a single variable, right? How to denote the space of functions that depend on two or three variables, say $(x,y)$ resp. $(x,y,z)$?

If I know the above mentioned spaces, then it should be possible to think of some operators in their duals, $X'$ and $Y'$. If I understand it correctly, the divergence operator should then be the operator to map operators from $Y'$ to $X'$.

So my actual question: how to denote the mentioned spaces, come up with a few operators in their duals $X'$ and $Y'$ (so some examples), and then show that the divergence indeed maps the operators from $Y'$ to $X'$?

• Adjoints are not duals, although they are closely related. See en.wikipedia.org/wiki/Hermitian_adjoint . Jul 16, 2012 at 15:30
• @QiaochuYuan: I thought that these operators were called dual or conjugate operators, and in the case of Hilbert spaces, adjoint operators? Anyway, is the rest of my question correct? Jul 16, 2012 at 15:44
• No. If $T : H_1 \to H_2$ is a bounded linear map between two Hilbert spaces, their dual is a map $T^{\ast} : H_2^{\ast} \to H_1^{\ast}$, but their adjoint is a map $T^{\dagger} : H_2 \to H_1$. The two do not have the same domain or the same range. Jul 16, 2012 at 15:48
• @QiaochuYuan: Ok, then let's say that both $X$ and $Y$ are Hilbert spaces. Can you then give an example of a space $X$ that is mapped to $Y$ by the gradient operator ($\nabla$)? Jul 16, 2012 at 16:14
• There are some subtleties here. The gradient operator acting on an appropriate subspace of the space of square-integrable functions $\mathbb{R}^n \to \mathbb{C}$ (say) is unbounded, so cannot be continuously extended to the whole space. One must talk about densely-defined operators instead and then talking about adjoints of such things rigorously is subtle. Jul 16, 2012 at 17:15