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An antiunitary operator between complex Hilbert spaces is a surjective antilinear map satisfying the displayed inner product identity. It preserves norms but conjugates scalars. In a chosen orthonormal basis it is $UK$, with $U$ a unitary operator and $K$ coefficientwise complex conjugation. Conversely any such $UK$ is antiunitary. Antilinearity by itself does not preserve norms.

The best mean-square linear predictor from a semi-infinite past is the orthogonal projection onto the closed linear span of past observations in the random-variable Hilbert space. For a causal invertible ARMA representation, past observations and past innovations generate the same closed span. The next linear innovation is orthogonal to it and is therefore the prediction error.

The best N-term approximation in an orthonormal basis retains $N$ coefficients of greatest absolute value. It minimizes the squared Hilbert space error over all index sets of size at most $N$. The selection makes the approximation nonlinear in the data, even though coefficient extraction is linear.

For a centered Gaussian measure on a real Hilbert space with injective covariance operator of a Gaussian measure $\Sigma$, its Cameron-Martin space is $E_{\mu }=\mathrm{Ran}{\Sigma }^{1/2}$, with inner product $⟨h,g{⟩}_{E_{\mu }}=⟨{\Sigma }^{-1/2}h,{\Sigma }^{-1/2}g⟩$. In an orthonormal basis satisfying $\Sigma e_j={\lambda }_j{e}_j$, it consists of exactly the vectors $h$ for which ${\sum }_j{h}_j^2/{\lambda }_j<\infty$. It is complete in this stronger norm even if its range is not closed in the ambient norm. Its importance is that it describes precisely the translations that preserve the null sets of the Gaussian measure, as in the Cameron-Martin theorem for a Gaussian measure. For generalized Gaussian white noise over $L^2$, the corresponding Cameron-Martin space is L2 space itself.

Let $\mathcal{G}$ be the first Gaussian chaos of a centered Gaussian random variable in a Banach space. By Fernique's theorem, $Sg=\mathbb{E}[Xg]$ is a Bochner integral in $B$. The map $S$ is injective, because $Sg=0$ implies $\mathbb{E}[g\ell (X)]=0$ for every continuous linear functional $\ell$, hence $g=0$. Its range becomes a Hilbert space under $⟨Sg,Sv{⟩}_H=\mathbb{E}[gv]$. The embedding in $B$ is continuous by Cauchy-Schwarz inequality. Its reproducing property is $⟨h,S\ell (X){⟩}_H=\ell (h)$. Thus this is the Reproducing-kernel Hilbert space of the random variable, including degenerate laws. For $h=Sg$, the scalar coordinate $\hat{h}(X)=g$ has variance $\Vert h{\Vert }_H^2$; it is generally a measurable linear coordinate rather than a continuous functional on $B$.

For a bounded linear operator $T$ between Hilbert spaces, its range is closed exactly when there is $b>0$ with $\Vert Tv\Vert \ge b\Vert v\Vert$ for all $v\in (\ker T{)}^{\perp }$. The restriction to this orthogonal complement is a bounded bijection onto the range, so the bounded inverse theorem proves necessity. Conversely, the bound makes preimages of a Cauchy sequence of image points a Cauchy sequence, proving closedness by completeness.

A vector subspace closed in the norm topology of a Hilbert space is complete in the inherited norm and inner product. Its orthogonal complement gives a decomposition of the ambient Hilbert space into an orthogonal direct sum. This closedness permits orthogonal projection and ensures limits used in range arguments remain in the subspace. Every finite-dimensional vector subspace is closed.

For a bounded operator $K$ between Hilbert spaces, compactness is equivalent to $f_n\rightharpoonup f$ implying $K{f}_n\to Kf$ in norm. The Uniform boundedness principle bounds a weakly convergent sequence. Relative compactness of its images and the unique possible weak limit then give norm convergence. Conversely, every bounded domain sequence has a weakly convergent subsequence, so the image of the unit ball is relatively compact.

For a Fredholm operator $L$ on a Hilbert space, invert its restriction from the kernel complement to its closed range, extending by zero on the range complement. The resulting bounded $B$ has $BL=I-P$ and $LB=I-Q$ with finite-rank projections. Adding a compact operator $K$ gives two-sided inverses modulo compact errors. Such a parametrix forces finite kernel, a lower bound on the kernel complement, closed range and finite adjoint kernel; the latter is the cokernel. Repeating with $-K$ proves the equivalence without assuming self-adjointness.

On the real Klein-Gordon field solution space with conserved symplectic form $\Omega$, a compatible real-linear map $J$ obeys $J^2=-I$, preserves $\Omega$, and makes $\mu (f,g)=\Omega (f,Jg)$ positive definite. Its $+i$ eigenspace in the complexified space has positive Klein-Gordon inner product with the convention $(u,v{)}_{KG}=i\Omega (\bar{u},v)$. It defines the one-particle Hilbert space and a free-field Fock vacuum.

For a nonnegative potential $V(x)\to \infty$ as $|x|\to \infty$, the confining-potential energy space isA local Poincare inequality controls the missing $L^2$ term, so this energy norm makes ${\Sigma }_V$ a Hilbert space.

The squared distance to a convex set is the infimal convolution of the set's indicator functional of a constraint set and the half-squared Hilbert space norm. Its convex conjugate is the sum of the half-squared norm and the support function of the set.

For proper lower semicontinuous convex functions on finite-dimensional Hilbert spaces, a nonempty saddle point set, and constant positive steps satisfying the displayed inequality, the primal-dual hybrid gradient method with extrapolation parameter one converges to a saddle point. The strict condition controls the bilinear coupling by the operator norm. It does not remove the need for existence of a saddle point. The fixed-step theorem and assumptions are recalled in Malitsky and Pock, A first-order primal-dual algorithm with linesearch, Section 1.

The covariant free-photon oscillator construction uses four polarization vectors and $[a^{\lambda },a^{{\lambda }^{\prime }†}]=-{\eta }^{\lambda {\lambda }^{\prime }}$. Starting from a positive Fock vacuum, it induces an indefinite Hermitian form on the multiparticle state space: the temporal oscillator has negative norm while the three spatial oscillators have positive norm. It is therefore not the physical positive Hilbert space. The Gupta-Bleuler null-state quotient selects a positive physical space with two transverse photon polarizations. Continuum momentum oscillators and their states are understood after smearing or finite-volume regularization.

A linear operator $T$ on a Hilbert space $H$ is densely defined when its operator domain is a dense subset of $H$. Density makes the adjoint operator unique wherever its defining inner-product identity has a solution.

For the Kac collision operator, expanding the squared Hilbert space norm gives $⟨F,(I-Q)F⟩=(4\pi (\genfrac{}{}{0ex}{}{N}{2}){)}^{-1}{\sum }_{i<j}{\int }_0^{2\pi }{\int }_{{\mathbb{R}}^N}|F(R_{ij,\theta }v)-F(v){|}^{2}dvd\theta$. The nonnegative form measures failure of invariance under the pair rotations. It vanishes precisely on the radial kernel of the Kac collision operator.

For a linear subspace $F$ of a Hilbert space,Thus $F=F^{\perp \perp }$ exactly when $F$ is closed.

The duality mapping takes a vector to its supporting dual functionals for half its squared norm. Equivalently $p\in \mathcal{J}(x)$ when $\Vert p{\Vert }_{\ast }=\Vert x\Vert$ and $⟨x,p⟩=\Vert x{\Vert }^2$. This follows from the dual-norm inequality and equality in the conjugate relation for squared norms. In a Hilbert space, the Riesz representation theorem identifies it with the identity; in a general Banach space it can be set-valued.

In a variational Hilbert space setting, a bounded operator $L$ is elliptic or uniformly coercive when $\mathrm{Re}⟨Lv,v⟩\ge \gamma \Vert v{\Vert }^2$ for a constant $\gamma >0$. On a real Hilbert space this condition depends on the symmetric part of $L$ and does not imply self-adjointness. It differs from the principal-symbol condition defining an elliptic differential operator.