TITLE: What is an intuitive view of adjoints? (version 2: functional analysis)
QUESTION [38 upvotes]: 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:

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.
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.

REPLY [4 votes]: I recommend this article, which gives a pretty damn good analogy between adjoints and complex conjugates.