Braketinlike

Braketinlike is a term used to describe a class of mathematical notations and structures that resemble the bra-ket notation of Dirac, used to express linear functionals and state vectors and their pairings. It is not a single formal theory, but a family of concepts used across physics and mathematics.

Foundations: It centers on dual pairing between a vector space V and its dual V*, with a pairing ⟨f|v⟩ giving a scalar. In standard quantum mechanics, |ψ⟩ is a ket in V and ⟨φ| is a bra in V*, with ⟨φ|ψ⟩ the scalar product. Braketinlike generalizes this to other contexts, including rigged Hilbert spaces, distributions, and category-theoretic dualities.

Notation and operations: The formalism emphasizes the flexibility of placement of bras and kets and the use of linear, antilinear, and antilinear in bras depending on conventions. It supports operators via outer products |ψ⟩⟨φ| and resolutions of identity ∑_i |i⟩⟨i|.

Applications and examples: In physics, it helps encode measurements, projections, and expectation values. In functional analysis, it provides a symbolic way to express bilinear forms and matrix elements. In teaching, it offers intuitive pictures of duality.

Etymology and status: The term Braketinlike is a neologism blending bra and ket with the suffix -inlike, used informally in literature and some educational contexts. It is not a standard formal theory but a descriptive shorthand for dual-pairing style notation.

---