TITLE: What are the benefits of writing vector inner products as $\langle u, v\rangle$ as opposed to $u^T v$?
QUESTION [103 upvotes]: In a lot of computational math, operations research, such as algorithm design for optimization problems and the like, authors like to use $$\langle \cdot, \cdot \rangle$$ as opposed to $$(\cdot)^T (\cdot)$$
Even when the space is clearly Euclidean and the operation is clearly the dot product. What is the benefit or advantage for doing so? Is it so that the notations generalize nicely to other spaces?
Update: Thank you for all the great answers! Will take a while to process...

REPLY [7 votes]: Lots of great answers so far, but I'll add another (hopefully at least good) answer: the notation $v^T u$ makes it somewhat difficult to speak of collections of bilinear pairings depending on a parameter.  Typical examples:

"Let $\langle \cdot, \cdot \rangle_i$ be a finite set of inner products on a vector space $V$"
"Let $\langle \cdot, \cdot \rangle_p$, $p \in M$, be a Riemannian metric on a manifold $M$"
"Let $\langle \cdot, \cdot \rangle_t$ be a continuously varying family of inner products on a Hilbert space $H$"

These are all difficult to express using the transpose notation.  The closest you can get is to write, for instance $v^T A_i u$ where $A_i$ is a family matrices, but particularly when one is speaking of continuously varying families of inner products you run into all sorts of difficult issues with coordinate systems, and it becomes very difficult to keep things straight.