| dbp:mathStatement
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- Let be a Hilbert space whose inner product is linear in its argument and antilinear in its second argument and let be the corresponding physics notation. For every continuous linear functional there exists a unique vector called the such that
Importantly for Hilbert spaces, is always located in the coordinate of the inner product.
Furthermore, the length of the representation vector is equal to the norm of the functional:
and is the unique vector with
It is also the unique element of minimum norm in ; that is to say, is the unique element of satisfying
Moreover, any non-zero can be written as (en)
- The is the injective linear operator isometry
The Riesz representation theorem states that this map is surjective when is complete and that its inverse is the bijective isometric antilinear isomorphism
Consequently, continuous linear functional on the Hilbert space can be written uniquely in the form where for every
The assignment can also be viewed as a bijective isometry into the anti-dual space of which is the complex conjugate vector space of the continuous dual space
The inner products on and are related by
and similarly,
The set satisfies and so when then can be interpreted as being the affine hyperplane that is parallel to the vector subspace and contains
For the physics notation for the functional is the bra where explicitly this means that which complements the ket notation defined by
In the mathematical treatment of quantum mechanics, the theorem can be seen as a justification for the popular bra–ket notation. The theorem says that, every bra has a corresponding ket and the latter is unique. (en)
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