I'm trying to find an explicit expression for the adjoint of a particular linear operator. Are there any very general techniques for doing so when we know very little about the setting we're working in? All I have is that is a linear operator and that it does have an adjoint. I know nothing about (inner product, dimension (possibly infinite), etc.).
The book doesn't give any examples like this, and the only one I've found online uses , so we can work with the inner product formulas explicitly to obtain the adjoint.
Yes, I do know that for an orthonormal basis . But since this only works for finite-dimensional vector spaces, I can't use it here. If I work on it for a while longer and don't come up with anything, I'll post here again and give more details. There's a particular aspect that's making it difficult.
I'm getting nowhere, so here's the actual problem:
"Let be an inner product space, and let . Define by for all . First prove that is linear. Then show that exists and find an explicit expression for it."
It's easy to show that is linear. It's the second part I'm having difficulty with. is the unique linear operator such that . In this case , so we need to show . What's throwing me off is that if we can get from the LHS to the RHS via manipulation by the usual useful properties of inner products, I see no way to get x into/out of the inner product and alone on the RHS. I'm probably over-complicating things, as I often seem to do. I would appreciate hints (and as always, only hints).