Wikipedia contributors (2026), Wikipedia — https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation
Key takeaways
From the reading, NOTE-855.
- Kets and bras. A ket |v⟩ is a vector of a complex vector space; a bra ⟨f| is a linear form on it. With an inner product antilinear in its first slot, each ket gives a bra, ⟨φ| = (φ, ·), and on a Hilbert space the article cites the Riesz theorem for every bra coming from a ket.
- Only on Hilbert space. On a Banach space or a space without topology bras and kets can still be written, but “the bracket does not have the meaning of an inner product, because the Riesz representation theorem does not apply”.
- Toolkit. Outer products |φ⟩⟨ψ| are rank-one operators and |ψ⟩⟨ψ| the projector onto a unit ψ; Hermitian conjugation reverses order and conjugates each factor; Σ_i |e_i⟩⟨e_i| = 𝕀 over a complete orthonormal system, with the continuous forms ∫|x⟩⟨x|dx = ∫|p⟩⟨p|dp = 𝕀.
- Improper kets. Position and momentum eigenkets are not in the Hilbert space; the article points to rigged Hilbert spaces and, less aptly, to the GNS construction.
- Date of origin. The reference the article gives is Dirac (1939), “A new notation for quantum mechanics”. The lead’s “monograph … 1930” is an unsourced edit of 2026-09-30.
Standing in the record
Filed on 2026-09-26 at the owner’s request, from a list grouped under the heading Riesz Representation Theorem, and read in full on 2026-10-14 in NOTE-855. It is reference material for notation. No THEORY, QUESTION or CLAIM cites it; NOTE-239 names it as the notation van Rijsbergen’s book (LIT-262) uses. The reading files no THEORY: the article adds no claim, and the one substantive point, that the bracket is an inner product only where Riesz holds, is the Riesz theorem itself (LIT-230). Cite it for conventions only, and for the notation’s date cite Dirac (1939), not its lead.