\(L^2\) is a Hilbert space with inner product \(\inner{f}{g}=\int f\bar g\). In a Hilbert space \(H\), every nonempty closed convex set has a unique element of least norm, so \(H=M\oplus M^\perp\) for every closed subspace \(M\); every orthonormal family \((e_i)\) satisfies Bessel’s inequality \(\sum_i\abs{\inner{x}{e_i}}^2\le\norm{x}^2\); and a complete orthonormal family \((e_i)_{i\in I}\) gives an isometric isomorphism \(H\cong\ell^2(I)\). The Riesz representation theorem: every bounded linear functional on \(H\) is \(x\mapsto\inner{x}{y}\) for a unique \(y\in H\). For \(1\le p<\infty\) on a \(\sigma\)-finite measure space, the dual of \(L^p\) is \(L^q\) with \(1/p+1/q=1\); the dual of \(L^\infty([0,1])\) strictly contains \(L^1([0,1])\). The Hahn–Banach, uniform boundedness, open mapping, and closed graph theorems apply to Banach spaces such as \(L^p\) and \(\ell^p\). The closed unit ball of an infinite-dimensional Hilbert space is not norm-compact, and every bounded sequence in a Hilbert space has a weakly convergent subsequence.
Hilbert space (Tag D-7QQUO)
A Hilbert space is an inner product space \((H,\inner{\cdot}{\cdot})\) over \(\RR\) or \(\CC\) that is a Banach space under the norm \(\norm{x}\coloneqq\sqrt{\inner{x}{x}}\).
Orthonormal sequence (Tag D-4IXAO)
Let \(H\) be an inner product space with inner product \(\inner{\cdot}{\cdot}\) and norm \(\norm{u}\coloneqq\sqrt{\inner{u}{u}}\), and let \(I\) be a countable index set. A family \((u_i)_{i\in I}\) of elements of \(H\) is orthonormal if
\(\inner{u_i}{u_j} = 0\) for all \(i,j\in I\) with \(i \neq j\), and
\(\norm{u_j} = 1\) for all \(j\in I\).
Bessel's inequality (Tag T-4BDE3)
Let \(\mch\) be a Hilbert space, let \((u_n)_{n\geq1}\) be an orthonormal sequence in \(\mch\), and let \(x\in\mch\). For every \(N\geq1\), \[ \norm{x-\sum_{n=1}^{N}\inner{x}{u_n}u_n}^{2}=\norm{x}^{2}-\sum_{n=1}^{N}\abs{\inner{x}{u_n}}^{2}, \] and consequently \[ \sum_{n=1}^{\infty}\abs{\inner{x}{u_n}}^{2} \leq\norm{x}^{2}. \]
Fix \(N\geq 1\) and put \(s_N\coloneqq\sum_{k=1}^{N}\inner{x}{u_k}u_k\). By orthonormality, \(\inner{x}{s_N} = \inner{s_N}{s_N} = \sum_{k=1}^{N}\abs{\inner{x}{u_k}}^2\), so \[ 0\leq\norm{x - s_N}^2 = \norm{x}^2 - 2\operatorname{Re}\inner{x}{s_N} + \norm{s_N}^2 = \norm{x}^2 - \sum_{k=1}^{N}\abs{\inner{x}{u_k}}^2. \] Letting \(N\to\infty\) gives the inequality.
The inequality extends to an orthonormal family \((u_i)_{i\in I}\) indexed by an arbitrary, possibly uncountable, set \(I\): for every \(x\in\mch\), \[ \sum_{i\in I}\abs{\inner{x}{u_i}}^2\coloneqq\sup_{\substack{F\subseteq I\\ F\text{ finite}}}\sum_{i\in F}\abs{\inner{x}{u_i}}^2\leq\norm{x}^2 . \] Consequently, for each \(x\) only countably many of the coefficients \(\inner{x}{u_i}\) are nonzero [@Fol13].
Complete families in a Hilbert space (Tag D-SLYE5)
Let \(H\) be a Hilbert space. A family \((u_{n})_{n\geq 1}\) of vectors in \(H\) is complete if the only \(x\in H\) with \(\inner{x}{u_{n}} = 0\) for all \(n\) is \(x = 0\).
Parseval's theorem for orthonormal families (Tag T-5AALA)
Let \(\mch\) be a Hilbert space and let \((u_n)_{n\in A}\) be a family in \(\mch\), indexed by a set \(A\), with \(\inner{u_n}{u_m}=0\) for \(n\neq m\) and \(\norm{u_n}=1\) for all \(n\). For each \(x\in\mch\) only countably many \(\inner{x}{u_n}\) are nonzero (T-4BDE3), and sums over \(A\) below are taken over those indices. The following are equivalent:
(Completeness) If \(x\in\mch\) and \(\inner{x}{u_n}=0\) for all \(n\in A\), then \(x=0\).
(Parseval’s identity) For every \(x\in\mch\), \[ \sum_{n\in A} \abs{ \inner{x}{u_n} }^2 = \norm{x}^2 . \]
(Expansion) For every \(x\in\mch\), \[ x = \sum_{n\in A} \inner{x}{u_n}u_n , \] with the series converging in norm in any enumeration of its nonzero terms; and if \(x=\sum_{n\in A}c_nu_n\) for scalars \(c_n\), only countably many nonzero, then \(c_n=\inner{x}{u_n}\) for all \(n\).
Riesz--Fischer theorem (Tag T-J3AN3)
Let \(\mathcal H\) be a Hilbert space, let \(U = (u_n)_{n\geq1}\) be an orthonormal sequence in \(\mathcal H\), and let \(\mathcal H_U\subseteq\mathcal H\) be the closure of the linear span of \(\theset{u_n\suchthat n\geq1}\). Then the coefficient map \[ \begin{aligned} \mathcal H_U &\longrightarrow \ell^2(\NN) \\ x &\longmapsto (\inner{x}{u_n})_{n\geq1} \end{aligned} \] is an isometric isomorphism. Equivalently, for every \((a_n)_{n\geq1}\in\ell^2(\NN)\) there is a unique \(x\in\mathcal H_U\) such that \[ a_n=\inner{x}{u_n}\quad\text{for all }n, \qquad\text{and then}\qquad \norm{x}^2=\sum_{n=1}^\infty\abs{a_n}^2. \]
In particular, if \(U\) is complete, so that \(\mathcal H_U=\mathcal H\), then the coefficient map \(\mathcal H\to\ell^2(\NN)\) is an isometric isomorphism.
For \(a=(a_n)\in\ell^2(\NN)\), set \[ x_N\coloneqq\sum_{n=1}^N a_nu_n. \] Orthonormality gives, for \(M<N\), \[ \norm{x_N-x_M}^2=\sum_{n=M+1}^N\abs{a_n}^2, \] so \((x_N)\) is Cauchy and converges, by completeness of \(\mathcal H\), to some \(x\), which lies in \(\mathcal H_U\) because each \(x_N\) lies in the span of \(U\). Continuity of the inner product yields \(\inner{x}{u_n}=a_n\) for every \(n\), and taking limits in \(\norm{x_N}^2=\sum_{n=1}^N\abs{a_n}^2\) gives \[ \norm{x}^2=\sum_{n=1}^\infty\abs{a_n}^2. \]
Conversely, let \(x\in\mathcal H_U\) and put \(a_n\coloneqq\inner{x}{u_n}\). Bessel’s inequality T-4BDE3 gives \(a\in\ell^2(\NN)\). Applying the construction above produces \(y\in\mathcal H_U\) with the same coefficients. Then \(x-y\in\mathcal H_U\) is orthogonal to every \(u_n\), hence to the span of \(U\) and therefore to its closure \(\mathcal H_U\); in particular \(x-y\) is orthogonal to itself, so \(x=y\). This proves bijectivity and the norm identity.
Linear functionals (Tag D-EPSKF)
Let \(X\) be a vector space over \(\CC\). A map \(L\colon X \to \CC\) is a linear functional if \[ L(\alpha x + y) = \alpha L(x) + L(y) \quad \text{for all } x,y\in X \text{ and } \alpha\in\CC. \]
Linear functionals are continuous if and only if bounded (Tag T-5IWCG)
Let \((X,\norm{\cdot})\) be a normed vector space over \(\CC\) and let \(L\colon X \to \CC\) be a linear functional. The following are equivalent:
\(L\) is continuous.
\(L\) is continuous at \(0\).
\(L\) is bounded: there exists \(c\geq 0\) such that \(\abs{L(x)} \leq c \norm{x}\) for all \(x\in X\).
Dual norm (Tag D-T4LOC)
Let \(X\neq\theset{0}\) be a normed vector space over \(\CC\), and let \(\dualof{X}\) be the space of continuous linear functionals \(X\to\CC\). For \(L \in \dualof{X}\), the dual norm or operator norm of \(L\) is \[ \norm{L}_{\dualof{X}} \coloneqq \sup_{ \substack{x\in X \\ \norm{x} = 1} } \abs{L(x)} = \sup_{ \substack{x\in X \\ \norm{x} \leq 1} } \abs{L(x)}. \]
Dual of a Hilbert space (Tag D-PQIQO)
Let \(H\) be a Hilbert space over \(\CC\). The dual of \(H\) is the set \(\dualof{H}\) of linear functionals \(L\colon H\to \CC\) that are continuous with respect to the norm of \(H\).
Riesz representation theorem for Hilbert spaces (Tag T-LDCZB)
Let \(H\) be a Hilbert space and let \(\Lambda\in \dualof{H}\) be a continuous linear functional on \(H\). Then there exists a unique \(y \in H\) such that \[ \Lambda(x) = \inner{x}{y} \quad\text{for all } x\in H . \]
Duals of $L^p$ spaces (Tag T-5BFVS)
Let \((X,\mcm,\mu)\) be a measure space, let \(1\leq p<\infty\), and let \(q\in(1,\infty]\) satisfy \(\frac1p+\frac1q=1\). If \(p=1\), assume that \(\mu\) is $\sigma$-finite. Then the map \[ L^q(X,\mu)\to \dualof{L^p(X,\mu)},\qquad g\mapsto\qty{f\mapsto\int_X fg\dmu}, \] is an isometric isomorphism onto the space \(\dualof{L^p(X,\mu)}\) of continuous linear functionals on \(L^p(X,\mu)\) with the dual norm [@Fol13].
Hahn--Banach theorem (Tag T-OW2QG)
Let \(\mathcal X\) be a real vector space and let \(p\colon \mathcal X\to\RR\) be sublinear, that is, \(p(x+y) \leq p(x) + p(y)\) and \(p(\lambda x) = \lambda p(x)\) for all \(x,y\in \mathcal X\) and all \(\lambda \geq 0\). Let \(\mathcal M \subseteq \mathcal X\) be a linear subspace and \(f\colon\mathcal M\to\RR\) a linear functional with \(f \leq p\) on \(\mathcal M\). Then there exists a linear functional \(F\colon\mathcal X\to\RR\) with \(\ro{F}{\mathcal M} = f\) and \(F \leq p\) on \(\mathcal X\).
Let \(\mathcal X\) be a complex vector space and let \(p\colon\mathcal X\to\RR\) be a seminorm, that is, \(p(x+y) \leq p(x) + p(y)\) and \(p(\lambda x) = \abs{\lambda} p(x)\) for all \(x,y\in \mathcal X\) and all \(\lambda \in\CC\). Let \(\mathcal M \subseteq \mathcal X\) be a linear subspace and \(f\colon\mathcal M\to\CC\) a linear functional with \(\abs f \leq p\) on \(\mathcal M\). Then there exists a linear functional \(F\colon\mathcal X\to\CC\) with \(\ro{F}{\mathcal M} = f\) and \(\abs F \leq p\) on \(\mathcal X\) [@Fol13].
The proof extends \(f\) one dimension at a time and applies Zorn’s lemma.
The extension \(F\) need not be unique. On \(\mathcal X=\RR^2\) let \(p(x)\coloneqq\abs{x_1}+\abs{x_2}\), \(\mathcal M\coloneqq\RR\times\theset{0}\), and \(f(x_1,0)\coloneqq x_1\). For every \(c\in[-1,1]\), the functional \(F_c(x)\coloneqq x_1+cx_2\) extends \(f\) and satisfies \(F_c\leq p\).
Let \(\mathcal X\) be a normed vector space over \(\RR\) or \(\CC\) with dual space \(\dualof{\mathcal X}\) of continuous linear functionals and dual norm.
If \(\mathcal M \subseteq \mathcal X\) is a closed linear subspace and \(x\in\mathcal X\setminus\mathcal M\), put \(\delta \coloneqq \inf_{y\in \mathcal M}\norm{x-y}\). Then there is \(f \in \dualof{\mathcal X}\) with \(\norm{f} = 1\), \(\ro f {\mathcal M} = 0\), and \(f(x) = \delta\).
For each \(x\in\mathcal X\) with \(x\neq 0\) there is \(f\in \dualof{\mathcal X}\) with \(\norm f = 1\) and \(f(x) = \norm x\). In particular, the functionals in \(\dualof{\mathcal X}\) separate the points of \(\mathcal X\).
The map \(\mathcal X \injects \mathcal X^{\vee\vee}\), \(x \mapsto \hat x\) with \(\hat x(f) \coloneqq f(x)\) for \(f\in\dualof{\mathcal X}\), is a linear isometry.
[@Fol13]
Uniform boundedness principle (Tag T-F2THV)
Let \(X\) be a Banach space, let \(Y\) be a normed vector space, and let \((T_\alpha)_{\alpha\in A}\) be a family of bounded linear operators \(T_\alpha\colon X\to Y\). Suppose the family is pointwise bounded: for every \(x\in X\) there exists \(C_x\geq0\) such that \[ \norm{T_{\alpha}x} \leq C_x \quad\text{for all } \alpha\in A . \] Then the family is uniformly bounded in operator norm: there exists \(C\geq0\) such that \[ \norm{T_\alpha x}\leq C\norm{x} \quad\text{for all } \alpha\in A \text{ and all } x\in X, \] that is, \(\sup_{\alpha\in A}\norm{T_\alpha}<\infty\) [@Fol13].
Open mapping theorem (Tag T-KQTPR)
Let \(X\) and \(Y\) be Banach spaces and let \(T\colon X\to Y\) be a continuous linear surjection. Then \(T\) is an open map: \(T(V)\) is open in \(Y\) for every open \(V\subseteq X\).
Closed graph theorem (Tag T-TTLXS)
Let \(X\) and \(Y\) be Banach spaces over the same field \(\RR\) or \(\CC\), and let \(T\colon X\to Y\) be a linear map whose graph \[ \Gamma(T)\coloneqq\theset{(x,Tx)\suchthat x\in X}\subseteq X\times Y \] is closed in \(X\times Y\) with the product topology. Then \(T\) is bounded: there exists \(C\geq0\) with \(\norm{Tx}_Y\leq C\norm{x}_X\) for all \(x\in X\).