Hilbert space

A Hilbert space is a complete inner-product space.

The closed unit ball of a Hilbert space is compact in norm exactly when the space is finite-dimensional. Finite-dimensional compactness follows from the Heine-Borel theorem. In infinite dimension, an orthonormal sequence has pairwise distance $\sqrt{2}$ and no convergent subsequence. Restricting a compact operator that equals a nonzero multiple of the identity on a subspace therefore forces that subspace to be finite-dimensional.

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.

A linear map $V:\mathcal{H}\to \mathcal{K}$ is an isometry when it preserves inner products, equivalently $V^{†}V=I$. It need not be surjective. It maps an orthonormal basis to an orthonormal family, and $\rho \mapsto V\rho V^{†}$ preserves the nonzero eigenvalues and the Von Neumann entropy of a density operator.

A linear isometry from a subspace of a finite-dimensional Hilbert space into the same ambient space extends to a unitary operator. Complete an orthonormal basis of the input subspace and its isometric image to full orthonormal bases, then map the first to the second. The equal complement dimensions make this possible. Orthogonality of the specified image columns, not normalization alone, is essential. In an infinite-dimensional space arbitrary isometries need not be surjective; the finite-dimensional same-space hypothesis cannot be dropped.

An N-term approximation in an orthonormal basis uses at most $N$ basis functions. For a chosen index set $\Lambda$, its optimal coefficients are the inner products with those functions, and its squared error is the sum of the omitted squared coefficients by the Parseval identity.

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 bounded function on an interval with finitely many jumps and piecewise $C^{\alpha }$ bounds, localized orthonormal wavelets with sufficient vanishing moments give squared best N-term approximation error $O(N^{-2\alpha })$. There are $O(2^j)$ smooth-region coefficients of size $O(2^{-j(\alpha +1/2)})$, and only $O(K)$ jump-crossing coefficients of size $O(2^{-j/2})$ per level. Retain all coefficients through level $J$ and the jump-crossing coefficients through $L=\lceil 2\alpha J\rceil$ when $\alpha >1/2$. This costs $O(2^J+KJ)$ and leaves squared tails $O(2^{-2\alpha J}+2^{-L})$. Choosing $2^J$ comparable to $N$ proves the rate by the Parseval identity.

A linear N-term approximation fixes its index set independently of the approximated function. Wavelet approximation normally orders by increasing resolution, whereas Fourier series approximation normally keeps the lowest frequencies. An arbitrary reordering can change or destroy a claimed convergence rate.

The Hilbert space completion of an inner-product space $V$ is a Hilbert space containing an isometric dense copy of $V$. It can be constructed from Cauchy sequences in $V$, identifying two sequences when the norm of their difference tends to zero.

Start with finite real linear combinations of $1$, $\cos (\lambda x)$ and $\sin (\lambda x)$ with arbitrary positive real frequencies, and use the inner product $⟨f,g{⟩}_M={\lim }_{R\to \infty }{R}^{-1}{\int }_{-R}^{R}fg$. Product-to-sum identities give existence of these averages and mutual orthogonality of distinct frequencies. The squared norm is $2a_0^2+{\sum }_{\lambda }(a_{\lambda }^2+b_{\lambda }^2)$. Its Hilbert space completion contains an uncountable orthonormal set, so is not a separable Hilbert space. This construction does not equip all locally square-integrable functions with an inner product: compactly supported nonzero functions have zero mean square, and other functions have divergent or nonexistent averages.

A Hilbert space is separable when it has a countable dense subset, equivalently a countable Hilbertian basis.

Every nonempty closed convex subset $C$ of a Hilbert space contains a unique point nearest to each $x$. A minimizing sequence is Cauchy by the parallelogram identity, and uniqueness follows by applying the same identity to two minimizers and their midpoint.

For a closed subspace $F$ of a Hilbert space,The closest point $P_{F}x$ gives the first component, and differentiating $\Vert x-P_{F}x-tu{\Vert }^2$ at zero shows that the remainder is orthogonal to every $u\in F$.

The orthogonal complement of a subset $F$ of an inner-product space is

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

The orthogonal projection $P_F$ onto a closed subspace $F$ sends $x$ to the component in the decomposition $x=P_{F}x+(I-P_F)x$ with $(I-P_F)x\in F^{\perp }$. It is a contraction: $\Vert P_{F}x\Vert \le \Vert x\Vert$.

For a nonzero closed subspace of a Hilbert space $V$ and a closed subspace of a Hilbert space $W$, this angle measures the worst loss under projection from $V$ to $W$. Its positive cosine is a uniform lower bound for $P_W{|}_V$. The order of the subspaces matters in general. This is distinct from the smallest angle between two subspaces, whose cosine is a supremum. For equal finite dimensions the cosine is the smallest singular value of the cross Gram matrix of orthonormal bases. When $V$ is zero, the infimum is over an empty unit sphere and does not define an angle in $[0,\pi /2]$; use a lower-bound formulation instead.

For closed subspaces of a Hilbert space, set $T=P_{V^{\perp }}{|}_W$. The two positive cosines bound $T$ and its adjoint operator $P_W{|}_{V^{\perp }}$ below. The first bound gives a closed range and injectivity; the second makes the range dense by image-kernel orthogonality for an adjoint. Thus $T$ is a bounded bijection, and $w=T^{-1}{P}_{V^{\perp }}f$ yields the unique direct sum decomposition $f=w+v$. In equal finite dimensions, a lower bound on $T$ alone suffices by the rank-nullity theorem.

For two finite-dimensional subspaces with projections $P,Q$, define $\cos \vartheta ={\sup }_{\Vert u\Vert =\Vert v\Vert =1}|⟨u,v⟩|=\Vert PQ\Vert$, with vectors drawn from the respective subspaces. The angle is zero when they intersect nontrivially. To measure separation after removing a common intersection, restrict both subspaces to its orthogonal complement. The resulting angle controls sums of positive operators in the Kitaev geometrical lemma.

Suppose positive operators $A,B$ have positive eigenvalues at least $\gamma$ and their nullspaces meet only at zero. If $\vartheta$ is their smallest angle between two subspaces, then $A+B\ge 2\gamma {\sin }^2(\vartheta /2)I$. Indeed $A+B\ge \gamma (2I-P-Q)$, while $(P+Q{)}^2\le (1+\cos \vartheta )(P+Q)$ gives $P+Q\le (1+\cos \vartheta )I$. With a common nullspace, apply the same proof on its orthogonal complement to bound the positive spectral gap.

For orthogonal projections $P,Q$ with $\Vert PQ\Vert \le q<1$, the identity $(PQ{)}^r=PQ(QPQ{)}^{r-1}$ and $\Vert QPQ\Vert =\Vert PQ{\Vert }^2$ imply $\Vert (PQ{)}^r\Vert \le q^{2r-1}$ for $r\ge 1$. When the projections have a common fixed subspace, first remove its orthogonal projection. This estimate is stronger than generic submultiplicativity and is useful for alternating local projection layers.

Every bounded linear functional on a Hilbert space is inner product with one unique vector, with equal norms.

For a nonzero bounded functional $f$ on a Hilbert space, choose nonzero $z\in (\ker f{)}^{\perp }$. Thenimplies $f(x)=⟨x,f(z)z/\Vert z{\Vert }^2⟩$ in the real case. Cauchy-Schwarz and evaluation on the representing vector give equality of the two norms.

For $T\in L(H)$, the adjoint $T^{\ast }$ is the unique bounded operator satisfyingRiesz representation applied for each $y$ proves existence; in an orthonormal basis its matrix is the conjugate transpose of the matrix of $T$.

When compositions and adjoints are defined on the common test domain, reversing products gives $[A,B{]}^{\ast }=[B^{\ast },A^{\ast }]$. In particular a self-adjoint Dolbeault Laplacian and the mutually adjoint Lefschetz operators satisfy $[{\Delta }_{\bar{\partial }},L{]}^{\ast }=[\Lambda ,{\Delta }_{\bar{\partial }}]$.

A densely defined operator $T$ on a Hilbert space is symmetric whenEquivalently, $T$ agrees with its adjoint operator on $D(T)$, while $D(T^{\ast })$ may be larger. A self-adjoint operator additionally requires equality of these operator domains.

For a linear map $T:V\to W$ between finite-dimensional inner product spaces,The first equality follows directly from the defining identity for the adjoint operator; the second follows by taking orthogonal complements. In an infinite-dimensional Hilbert space, the second equality generally requires closure of the image.

A bounded operator $T$ on a Hilbert space is invertible exactly when both $T$ and $T^{\ast }$ are bounded below. A lower bound on $T$ makes it injective with closed range, while a lower bound on $T^{\ast }$ gives $\ker T^{\ast }=0$ and hence dense range throughClosed dense range is the whole space, and the lower bound controls the inverse.

Hermitian conjugation takes the complex conjugate and reverses the order of a product of operators: $(AB{)}^{†}=B^{†}{A}^{†}$. An operator satisfying $A^{†}=A$ is Hermitian.

The formal adjoint of a differential operator $L$ is characterized by moving $L$ between factors under an integral by integration by parts, while discarding boundary terms for compactly supported functions.

For an order-$m$ operator with smooth coefficients on a bounded piecewise smooth domain, integration by parts has no boundary remainder when at each boundary point either the entire boundary jet of $u$ through order $m-1$ vanishes or the entire such jet of $v$ vanishes. Every boundary summand pairs derivatives of the two functions of order at most $m-1$. It is insufficient to require only that, for each same multi-index $\beta$, one of ${\partial }^{\beta }u$ and ${\partial }^{\beta }v$ vanish: for $P={\partial }_x^2$, $u=x(a-x)\chi (y)$ and $v=1$ on a rectangle, all those same-index products vanish on the boundary but $\int (Pu)v=-2a\int \chi \ne 0$.

A bounded operator is Hermitian, or self-adjoint, when $T=T^{\ast }$.

For a complex $n\times n$ matrix $B$, the real linear map $A\mapsto BA{B}^{\ast }$ on the $n^2$-dimensional real vector space of Hermitian matrices has determinant $|\det B{|}^{2n}$. A real basis of Hermitian matrices is also a complex basis of all matrices, so the matrix of this restriction equals the matrix of the complex-linear congruence map. Left and right multiplication separately have determinants $(\det B{)}^n$ and $(\overline{\det B}{)}^n$.

A bounded Hermitian operator $T$ on a Hilbert space is positive when $⟨Tx,x⟩\ge 0$ for every $x$. Its spectrum is contained in $[0,\infty )$.

A symmetric operator $L$ on a real Hilbert space is strictly positive definite when $⟨Lv,v⟩>0$ for every nonzero $v$ in its operator domain. Strict positivity proves uniqueness of a solution of $Lu=f$, but in infinite dimension it need not prove existence for every $f$. A uniformly positive definite symmetric operator additionally has a positive lower bound independent of $v$. Symmetry must be stated separately when positivity is defined only by a real quadratic expression: adding a skew-symmetric matrix changes the operator without changing that expression.

For a symmetric bounded bilinear form $B$ that is strictly positive on nonzero vectors, and a bounded linear functional $\ell$, set $I(v)=B(v,v)-2\ell (v)$. The first variation is $DI(v)[w]=2(B(v,w)-\ell (w))$. A weak solution $u$ of $B(u,w)=\ell (w)$ satisfies $I(u+w)-I(u)=B(w,w)$, so it is the unique minimizer. Conversely, any minimizer solves the weak equation. A coercive bilinear form gives existence by the Lax-Milgram theorem; strict positivity alone does not.

For a bounded operator on a real Hilbert space, $⟨Lv,v⟩=⟨Sv,v⟩$ with $S=(L+L^{\ast })/2$. Consequently the Euler-Lagrange equation of $⟨Lv,v⟩-2⟨f,v⟩$ is $Sv=f$. It is $Lv=f$ only when $L$ is self-adjoint or when the relevant solution also annihilates the skew part. For example $L=\left(\begin{array}{cc}1& -1\\ 1& 1\end{array}\right)$ has positive quadratic form $\Vert v{\Vert }^2$, but that form contains no information about its skew part.

A positive definite symmetric operator is uniformly positive definite if $⟨Lv,v⟩\ge \gamma \Vert v{\Vert }^2$ for some $\gamma >0$. For a bounded linear operator defined on a whole Hilbert space, this is a symmetric coercive operator, and the Lax-Milgram theorem supplies a unique solution to $Lu=f$ for every $f$. For a differential operator, the corresponding bounded coercive form is used on its form space.

A positive contraction is a positive operator whose operator norm is at most one. Its spectrum lies in $[0,1]$, so the spectral theorem for normal operators gives $M^2\le M$. Positive contractions are the effects in a POVM and occur in the pure-state gentle measurement bound.

The support of a positive operator is the orthogonal complement of its kernel. In finite dimension it is the span of eigenvectors with positive eigenvalues, equivalently the operator's image. Quantum relative entropy is finite only when the first state's support lies in the reference state's support.

Every bounded positive operator $T$ has a unique positive operator $T^{1/2}$ satisfying $(T^{1/2}{)}^2=T$. The spectral theorem for normal operators on a separable Hilbert space constructs it by applying the scalar square-root function to the spectrum.

The spectral theorem for normal operators gives a unique decompositionwhere $A_{+},A_{-}\ge 0$ and their supports are orthogonal. They are called the positive and negative parts of $A$. If $\mathrm{Tr}A=0$, then $\mathrm{Tr}{A}_{+}=\mathrm{Tr}{A}_{-}=\Vert A{\Vert }_1/2$.

The negative part of a Hermitian operator is $X_{-}={\sum }_i\max (-{\lambda }_i,0)P_i=(|X|-X)/2$. In this convention $X_{-}$ is itself a positive operator, and $X=X_{+}-X_{-}$. Its support is orthogonal to that of the positive part of a Hermitian operator.

For a Hermitian operator with spectral decomposition $X={\sum }_i{\lambda }_i{P}_i$, its positive part is $X_{+}={\sum }_i\max ({\lambda }_i,0)P_i=(|X|+X)/2$. It is a positive operator supported on the positive spectral subspace. Together with the negative part of a Hermitian operator it satisfies $X=X_{+}-X_{-}$ and $|X|=X_{+}+X_{-}$.

For Hermitian operators $A$ and $B$, one writes $A\le B$ when $B-A$ is positive semidefinite. This partial order is also called the semidefinite order.

For bounded positive self-adjoint operators, $0\le A\le B$ implies $A^{1/2}\le B^{1/2}$. Thus the nonnegative square-root function is operator monotone.

For a subspace $U$ of a finite-dimensional inner-product space,This follows directly by moving $T$ across the inner product and using $(U^{\perp }{)}^{\perp }=U$.

An operator is normal when $T{T}^{\ast }=T^{\ast }T$. Equivalently,for every $x$; the converse follows by polarizing the quadratic form of the self-adjoint commutator $T^{\ast }T-T{T}^{\ast }$.

For a normal bounded operator, $\sigma (T)={\sigma }_{\mathrm{ap}}(T)$. If $T-\lambda I$ is bounded below, its range is closed; normality makes its adjoint injective, so the range is also dense and the operator is invertible.

Every normal operator on a finite-dimensional complex inner-product space has an orthonormal basis of eigenvectors. An eigenvector is also an eigenvector of the adjoint with conjugate eigenvalue, so its orthogonal complement reduces the operator and induction applies.

For a Hermitian matrix with unitary diagonalization $A=U\mathrm{diag}({\lambda }_i)U^{\ast }$, define $f(A)=U\mathrm{diag}(f({\lambda }_i))U^{\ast }$ for a function on its finite spectrum. This acts on eigenvalues, unlike an entrywise matrix function $f[A]$. For example, the spectral square of a matrix is its ordinary matrix product with itself, whereas its entrywise square is a Hadamard power.

A normal operator on a separable Hilbert space has a unique projection-valued spectral measure $E$ such that $A=\int zdE(z)$, with the natural square-integrability domain in the unbounded case.

A projection-valued measure $E$ assigns an orthogonal projection $E(B)$ to each measurable set, with $E(\varnothing )=0$, $E(X)=I$, $E(B\cap C)=E(B)E(C)$, and countable additivity in the strong operator topology on pairwise disjoint sets.

For a unital star-representation $\pi :C(K)\to \mathcal{B}(H)$, the Riesz-Markov-Kakutani representation theorem gives regular measures ${\mu }_{x,y}$ representing $f\mapsto ⟨\pi (f)x,y⟩$. Define $P(E)$ by $⟨P(E)x,y⟩={\mu }_{x,y}(E)$. Positivity and ${\mu }_{x,x}(K)=\Vert x{\Vert }^2$ make these positive contractions. The identities ${\mu }_{\pi (f)x,y}=f{\mu }_{x,y}$ imply commutation with $\pi (f)$ and ${\mu }_{P(E)x,y}=1_E{\mu }_{x,y}$; hence $P(F)P(E)=P(F\cap E)$. Thus each $P(E)$ is an orthogonal projection. Scalar countable additivity and orthogonality give countable additivity in the strong operator topology, constructing the normalized regular projection-valued measure.

If $A$ is a commutative C-star algebra of bounded operators on a Hilbert space containing its identity, there is a unique regular projection-valued measure $E$ on the character space ${\Phi }_A$ such that $E({\Phi }_A)=I$ and $a={\int }_{{\Phi }_A}\hat{a}dE$ for every $a\in A$, where $\hat{a}$ is the Gelfand transform.

For a faithful representation of $C(K)$ by integration against a regular projection-valued measure $P$, every nonempty open subset $U$ satisfies $P(U)\ne 0$. Choose a nonzero continuous function supported inside $U$, using compact Hausdorff normality. If $P(U)=0$, its spectral integral vanishes, contradicting faithfulness. Regularity is understood through the scalar spectral measures.

A spectral projector is the value $E(B)$ of the projection-valued measure of a normal operator on a measurable part $B$ of its spectrum. In finite dimension, the spectral projector onto an eigenspace extracts the corresponding eigenvector component.

For vectors $f,g$, the scalar spectral measure is ${\mu }_{f,g}(S)=⟨E(S)f,g⟩$. Functional calculus satisfies $⟨h(A)f,g⟩=\int hd{\mu }_{f,g}$.

For a bounded normal operator $T$ with spectral measure of a normal operator $P$, any projection $P(E)$ commutes with $T$ and $T^{\ast }$ by the Borel functional calculus for a normal operator. Its range is a closed reducing subspace. If the spectrum has two points, choose disjoint nonempty relative open sets around them. Full support of a faithful spectral measure makes both projections nonzero and their product zero, so either range is nonzero and proper, in particular an invariant subspace.

For a projection-valued measure $E$ and vectors $v,w$, the scalar spectral measure is the complex measure ${\mu }_{v,w}(B)=⟨E(B)v,w⟩$. The positive measure ${\mu }_v={\mu }_{v,v}$ has total mass $\Vert v{\Vert }^2$.

Scalar spectral measures ${\mu }_{v,w}^{(n)}$ converge weakly to ${\mu }_{v,w}$ when $\int fd{\mu }_{v,w}^{(n)}\to \int fd{\mu }_{v,w}$ for every bounded continuous function $f$. Uniform second-moment bounds supply the tightness needed to pass from polynomial tests to all such $f$.

Stone's formula recovers spectral projections of a self-adjoint operator from the jump of its resolvent across the real axis, with half weight at interval endpoints.

A sequence $x_n$ converges weakly to $x$ when $⟨x_n,y⟩\to ⟨x,y⟩$ for every $y$ in the Hilbert space.

If $x_n\rightharpoonup x$ in a Hilbert space, some subsequence $(x_{n_k})$ has Cesaro means converging in norm to $x$. For $x=0$, choose the subsequence so successive vectors have summably small inner products; expansion of the squared norm of its averages then proves convergence.

For a Hilbertian basis $(e_i)$, a sequence $x_n$ converges weakly if and only if $(x_n)$ is norm bounded and every coordinate sequence $⟨x_n,e_i⟩$ converges.

Every bounded sequence in a separable Hilbert space has a weakly convergent subsequence. Successive subsequences make each basis coordinate converge, and a diagonal subsequence converges in every coordinate; the coordinate criterion for weak convergence in a separable Hilbert space finishes the proof.

If $x_n\rightharpoonup x$ in a Hilbert space, thenFor $x\ne 0$, take the inner product with $x$ and apply Cauchy-Schwarz before passing to the lower limit.

In a Hilbert space, weak convergence $x_n\rightharpoonup x$ together with $\Vert x_n\Vert \to \Vert x\Vert$ implies strong convergence $x_n\to x$.

Suppose $x_n\rightharpoonup x$ in a separable Hilbert space with Hilbertian basis $(e_i)$. Then $x_n\to x$ in norm exactly whenThe condition prevents norm from escaping to successively higher basis coordinates.

Every orthonormal sequence in a Hilbert space converges weakly to zero, by Bessel inequality, but no subsequence converges strongly because distinct terms remain distance $\sqrt{2}$ apart.

If $x_n$ converges weakly to $x$ in a Banach space, there are convex combinations of each tail $\{x_n,x_{n+1},\dots \}$ that converge in norm to $x$.

The weak closure and norm closure of a convex subset of a real or complex normed space coincide. If a point lies outside the norm closure, the Hahn-Banach separation theorem supplies a weakly continuous linear functional that separates it from the convex set.

A norm-closed convex subset of a Banach space contains every weak limit of its sequences. Apply Mazur lemma to obtain norm-convergent convex combinations that remain in the set.

An orthonormal sequence $(e_n)$ satisfies $⟨e_n,e_m⟩=0$ for $n\ne m$ and $\Vert e_n\Vert =1$.

A Hilbertian basis is a complete orthonormal family: its closed linear span is the whole Hilbert space, equivalently every vector is the norm-convergent sum of its Fourier coefficients against the basis.

For a Hilbertian basis $(e_i)$,

For an orthonormal sequence $(e_n)$ in a Hilbert space,