After realising that I don't have an intuitive understanding of adjoint **functors**, I then realised that I don't have an intuitive understanding of adjoint **linear transformations**!


Again, I can use 'em, compute 'em, and convolute 'em, but I have no real intuition as to what is going on. My best attempts (when teaching them) were:


1. The adjoint allows us to shift stuff from one side of the inner product to the other, thus, in a fashion, moving it out of the way while we do something and then moving it back again.
2. Nice behaviour with respect to the adjoint (say, normal or unitary) translates into nice behaviour with respect to the inner product.


But neither of those is really about what the adjoint **is**, just what it does.



What is an intuitive view of adjoints? (version 2: functional analysis)
Just to add to Yemon's answer, in the case of inner product spaces I think it may be helpful to understand the case of operators of finite rank. So let's start with the shockingly simple case where $\eta$ is a single vector in the inner product space $H$. I like to identify this with the linear map $t\mapsto t\eta$ from $\mathbb{C}$ to $H$, whose adjoint is the map $\xi\mapsto\langle\xi,\eta\rangle$ – in other words, just the functional associated with $\xi$. Going the other way (assuming $H$ is complete), the adjoint of a linear functional “is”, of course, just the vector provided to us by the Riesz representation theorem. To physicists, who write the inner product backwards (with good reason, btw) we are mere looking at the bra and ket version of the same vector, so $\langle\eta|$ and $|\eta\rangle$ will be each others' adjoints.


Next, you can tackle rank one operators between Hilbert spaces, which will be of the form $\xi\mapsto\langle\eta,\xi\rangle\zeta$ for fixed vectors $\eta$ and $\zeta$. And once you have a fairly good idea how that works, it's a short way to adding a finite number of these to get any finite rank operator.


Admittedly, from here to infinite rank is something of a stretch, but I think it might help anyhow.


