Hilbert Space - Properties - Duality

Duality

The dual space H* is the space of all continuous linear functions from the space H into the base field. It carries a natural norm, defined by

This norm satisfies the parallelogram law, and so the dual space is also an inner product space. The dual space is also complete, and so it is a Hilbert space in its own right.

The Riesz representation theorem affords a convenient description of the dual. To every element u of H, there is a unique element φu of H*, defined by

The mapping is an antilinear mapping from H to H*. The Riesz representation theorem states that this mapping is an antilinear isomorphism. Thus to every element φ of the dual H* there exists one and only one uφ in H such that

for all xH. The inner product on the dual space H* satisfies

The reversal of order on the right-hand side restores linearity in φ from the antilinearity of uφ. In the real case, the antilinear isomorphism from H to its dual is actually an isomorphism, and so real Hilbert spaces are naturally isomorphic to their own duals.

The representing vector uφ is obtained in the following way. When φ ≠ 0, the kernel F = Ker(φ) is a closed vector subspace of H, not equal to H, hence there exists a non-zero vector v orthogonal to F. The vector u is a suitable scalar multiple λv of v. The requirement that φ(v) = ⟨v, u⟩ yields

This correspondence φu is exploited by the bra-ket notation popular in physics. It is common in physics to assume that the inner product, denoted by ⟨x|y⟩, is linear on the right,

The result ⟨x|y⟩ can be seen as the action of the linear functional ⟨x| (the bra) on the vector |y⟩ (the ket).

The Riesz representation theorem relies fundamentally not just on the presence of an inner product, but also on the completeness of the space. In fact, the theorem implies that the topological dual of any inner product space can be identified with its completion. An immediate consequence of the Riesz representation theorem is also that a Hilbert space H is reflexive, meaning that the natural map from H into its double dual space is an isomorphism.

Read more about this topic:  Hilbert Space, Properties

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