Symplectic Hilbert Space (K,B)

A weak symplectic Hilbert space is a real Hilbert space $K$ equipped with a continuous, skew-symmetric bilinear form $B:K\times K\to\mathbb R$ such that

B(x,y)=0\ \text{for every }y\in K\quad\Longrightarrow\quad x=0.

Thus $B$ is a continuous symplectic form in the algebraic sense.

Strong versus weak nondegeneracy

The form defines a bounded operator

B^\flat:K\longrightarrow K^*,\qquad B^\flat(x)=B(x,\mathord{-}).

Weak nondegeneracy says that $B^\flat$ is injective. The symplectic form is strong if $B^\flat$ is an isomorphism of Banach spaces. These conditions coincide in finite dimensions but not in infinite dimensions. Authors who use “symplectic Hilbert space” without a qualifier may mean either the weak or the strong notion, so the convention must be stated.

Shale's setting

If $H$ is a complex Hilbert space, its underlying real Hilbert space $H_{\mathbb R}$ becomes symplectic with

B(z_1,z_2)=\operatorname{Im}\langle z_1,z_2\rangle.

This form is strong: multiplication by $i$ , together with the real Riesz isomorphism, identifies $H_{\mathbb R}$ continuously with its dual. Its bounded real-linear symplectic automorphisms form $\operatorname{Sp}(K)$ .