Distances in Functional Analysis

Abstract

Functional Analysis is the branch of Mathematics concerned with the study of spaces of functions. This usage of the word functional goes back to the calculus of variations which studies functions whose argument is a function. In the modern view, Functional Analysis is seen as the study of complete normed vector spaces, i.e., Banach spaces.

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Functional Analysis is the branch of Mathematics concerned with the study of spaces of functions. This usage of the word functional goes back to the calculus of variations which studies functions whose argument is a function. In the modern view, Functional Analysis is seen as the study of complete normed vector spaces, i.e., Banach spaces.

For any real number p ≥ 1, an example of a Banach space is given by L _p -space of all Lebesgue-measurable functions whose absolute value’s p-th power has finite integral.

A Hilbert space is a Banach space in which the norm arises from an inner product. Also, in Functional Analysis are considered continuous linear operators defined on Banach and Hilbert spaces.

1 Metrics on Function Spaces

Let $I \subset \mathbb{R}$ be an open interval (i.e., a nonempty connected open set) in $\mathbb{R}$. A real function $f: I \rightarrow \mathbb{R}$ is called real analytic on I if it agrees with its Taylor series in an open neighborhood $U_{x_{0}}$ of every point x ₀ ∈ I: $f(x) =\sum _{ n=0}^{\infty }\frac{f^{(n)}(x_{ 0})} {n!} (x - x_{0})^{n}$ for any $x \in U_{x_{0}}$. Let $D \subset \mathbb{C}$ be a domain (i.e., a convex open set) in $\mathbb{C}$.

A complex function $f: D \rightarrow \mathbb{C}$ is called complex analytic (or, simply, analytic) on D if it agrees with its Taylor series in an open neighborhood of every point z ₀ ∈ D. A complex function f is analytic on D if and only if it is holomorphic on D, i.e., if it has a complex derivative $f^{^{{\prime}} }(z_{0}) =\lim _{z\rightarrow z_{0}} \frac{f(z)-f(z_{0})} {z-z_{0}}$ at every point z ₀ ∈ D.

• Integral metric

  The integral metric is the L ₁ -metric on the set C _[a, b] of all continuous real (complex) functions on a given segment [a, b] defined by

  $$\displaystyle{\int _{a}^{b}\vert f(x) - g(x)\vert \mathit{dx}.}$$

  The corresponding metric space is abbreviated by C _[a, b] ¹. It is a Banach space.

  In general, for any compact topological space X, the integral metric is defined on the set of all continuous functions $f: X \rightarrow \mathbb{R}$ ($\mathbb{C}$) by ∫ _X  | f(x) − g(x) | dx.

• Uniform metric

  The uniform metric (or sup metric ) is the L _∞ -metric on the set C _[a, b] of all real (complex) continuous functions on a given segment [a, b] defined by

  $$\displaystyle{\sup _{x\in [a,b]}\vert f(x) - g(x)\vert.}$$

  The corresponding metric space is abbreviated by C _[a, b] ^∞. It is a Banach space.

  A generalization of C _[a, b] ^∞ is the space of continuous functions C(X), i.e., a metric space on the set of all continuous (more generally, bounded) functions $f: X \rightarrow \mathbb{C}$ of a topological space X with the L _∞ -metric sup _x ∈ X  | f(x) − g(x) | .

  In the case of the metric space C(X, Y ) of continuous (more generally, bounded) functions f: X → Y from one metric compactum (X, d _X ) to another (Y, d _Y ), the sup metric between two functions f, g ∈ C(X, Y ) is defined by $\sup _{x\in X}d_{Y }(f(x),g(x))$.

  The metric space C _[a, b] ^∞, as well as the metric space C _[a, b] ¹, are two of the most important cases of the metric space C _[a, b] ^p, 1 ≤ p ≤ ∞, on the set C _[a, b] with the L _p -metric $(\int _{a}^{b}\vert f(x) - g(x)\vert ^{p}\mathit{dx})^{\frac{1} {p} }$. The space C _[a, b] ^p is an example of an L _p -space.

• Dogkeeper distance

  Given a metric space (X, d), the dogkeeper distance is a metric on the set of all functions f: [0, 1] → X, defined by

  $$\displaystyle{\inf _{\sigma }\sup _{t\in [0,1]}d(f(t),g(\sigma (t))),}$$

  where σ: [0, 1] → [0, 1] is a continuous, monotone increasing function such that σ(0) = 0, σ(1) = 1. This metric is a special case of the Fréchet metric.

  For the case, when (X, d) is Euclidean space $\mathbb{R}^{n}$, this metric is the original (1906) Fréchet distance between parametric curves $f,g: [0,1] \rightarrow \mathbb{R}^{n}$. This distance can be seen as the length of the shortest leash that is sufficient for the man and the dog to walk their paths f and g from start to end. For example, the Fréchet distance between two concentric circles of radius r ₁ and r ₂ respectively is | r ₁ − r ₂ | .

  The discrete Fréchet distance (or coupling distance, Eiter and Mannila, 1994) is an approximation of the Fréchet metric for polygonal curves f and g. It considers only positions of the leash where its endpoints are located at vertices of f and g. So, this distance is the minimum, over all order-preserving pairings of vertices in f and g, of the maximal Euclidean distance between paired vertices.

• Bohr metric

  Let $\mathbb{R}$ be a metric space with a metric ρ. A continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ is called almost periodic if, for every ε > 0, there exists l = l(ε) > 0 such that every interval [t ₀, t ₀ + l(ε)] contains at least one number τ for which ρ(f(t), f(t +τ)) < ε for $-\infty < t < +\infty$.

  The Bohr metric is the norm metric | | f − g | | on the set AP of all almost periodic functions defined by the norm

  $$\displaystyle{\vert \vert f\vert \vert =\sup _{-\infty <t<+\infty }\vert f(t)\vert.}$$

  It makes AP a Banach space. Some generalizations of almost periodic functions were obtained using other norms; cf. Stepanov distance, Weyl distance, Besicovitch distance and Bochner metric.

• Stepanov distance

  The Stepanov distance is a distance on the set of all measurable functions $f: \mathbb{R} \rightarrow \mathbb{C}$ with summable p-th power on each bounded integral, defined by

  $$\displaystyle{\sup _{x\in \mathbb{R}}\left (\frac{1} {l} \int _{x}^{x+l}\vert f(x) - g(x)\vert ^{p}\mathit{dx}\right )^{1/p}.}$$

  The Weyl distance is a distance on the same set defined by

  $$\displaystyle{\lim _{l\rightarrow \infty }\sup _{x\in \mathbb{R}}\left (\frac{1} {l} \int _{x}^{x+l}\vert f(x) - g(x)\vert ^{p}\mathit{dx}\right )^{1/p}.}$$

• Besicovitch distance

  The Besicovitch distance is a distance on the set of all measurable functions $f: \mathbb{R} \rightarrow \mathbb{C}$ with summable p-th power on each bounded integral defined by

  $$\displaystyle{\left (\overline{\lim }_{T\rightarrow \infty } \frac{1} {2T}\int _{-T}^{T}\vert f(x) - g(x)\vert ^{p}\mathit{dx}\right )^{1/p}.}$$

  The generalized Besicovitch almost periodic functions correspond to this distance.

• Bochner metric

  Given a measure space $(\Omega,\mathcal{A},\mu )$, a Banach space (V, | | . | |  _V ), and 1 ≤ p ≤ ∞, the Bochner space (or Lebesgue–Bochner space) $L^{p}(\Omega,V )$ is the set of all measurable functions $f: \Omega \rightarrow V$ such that $\vert \vert f\vert \vert _{L^{p}(\Omega,V )} \leq \infty$.

  Here the Bochner norm $\vert \vert f\vert \vert _{L^{p}(\Omega,V )}$ is defined by $(\int _{\Omega }\vert \vert f(\omega )\vert \vert _{V }^{p}d\mu (\omega ))^{\frac{1} {p} }$ for 1 ≤ p < ∞, and, for p = ∞, by $\mathrm{ess}\sup _{\omega \in \Omega }\vert \vert f(\omega )\vert \vert _{V }$.

• Bergman p -metric

  Given 1 ≤ p < ∞, let $L_{p}(\Delta )$ be the L _p -space of Lebesgue measurable functions f on the unit disk $\Delta =\{ z \in \mathbb{C}: \vert z\vert < 1\}$ with $\vert \vert f\vert \vert _{p} = \left (\int _{\Delta }\vert f(z)\vert ^{p}\mu (\mathit{dz})\right )^{\frac{1} {p} } < \infty$.

  The Bergman space $L_{p}^{a}(\Delta )$ is the subspace of $L_{p}(\Delta )$ consisting of analytic functions, and the Bergman p -metric is the L _p -metric on $L_{p}^{a}(\Delta )$ (cf. Bergman metric in Chap. 7). Any Bergman space is a Banach space.

• Bloch metric

  The Bloch space B on the unit disk $\Delta =\{ z \in \mathbb{C}: \vert z\vert < 1\}$ is the set of all analytic functions f on $\Delta$ such that $\vert \vert f\vert \vert _{B} =\sup _{z\in \Delta }(1 -\vert z\vert ^{2})\vert f^{^{{\prime}} }(z)\vert < \infty$. Using the complete seminorm | | . | |  _B , a norm on B is defined by

  $$\displaystyle{\vert \vert f\vert \vert = \vert f(0)\vert + \vert \vert f\vert \vert _{B}.}$$

  The Bloch metric is the norm metric | | f − g | | on B. It makes B a Banach space.

• Besov metric

  Given 1 < p < ∞, the Besov space B _p on the unit disk $\Delta =\{ z \in \mathbb{C}: \vert z\vert < 1\}$ is the set of all analytic functions f in $\Delta$ such that $\vert \vert f\vert \vert _{B_{p}} = \left (\int _{\Delta }(1 -\vert z\vert ^{2})^{p}\vert f^{^{{\prime}} }(z)\vert ^{p}d\lambda (z)\right )^{\frac{1} {p} } < \infty$, where $d\lambda (z) = \frac{\mu (\mathit{dz})} {(1-\vert z\vert ^{2})^{2}}$ is the Möbius invariant measure on $\Delta$. Using the complete seminorm $\vert \vert.\vert \vert _{B_{p}}$, the Besov norm on B _p is defined by

  $$\displaystyle{\vert \vert f\vert \vert = \vert f(0)\vert + \vert \vert f\vert \vert _{B_{p}}.}$$

  The Besov metric is the norm metric | | f − g | | on B _p .

  It makes B _p a Banach space. The set B ₂ is the classical Dirichlet space of functions analytic on $\Delta$ with square integrable derivative, equipped with the Dirichlet metric . The Bloch space B can be considered as B _∞ .

• Hardy metric

  Given 1 ≤ p < ∞, the Hardy space $H^{p}(\Delta )$ is the class of functions, analytic on the unit disk $\Delta =\{ z \in \mathbb{C}: \vert z\vert < 1\}$, and satisfying the following growth condition for the Hardy norm $\vert \vert.\vert \vert _{H^{p}}$:

  $$\displaystyle{\vert \vert f\vert \vert _{H^{p}(\Delta )} =\sup _{0<r<1}\left (\frac{1} {2\pi }\int _{0}^{2\pi }\vert f(re^{i\theta })\vert ^{p}d\theta \right )^{\frac{1} {p} } < \infty.}$$

  The Hardy metric is the norm metric $\vert \vert f - g\vert \vert _{H^{p}(\Delta )}$ on $H^{p}(\Delta )$. It makes $H^{p}(\Delta )$ a Banach space.

  In Complex Analysis, the Hardy spaces are analogs of the L _p -spaces of Functional Analysis. Such spaces are applied in Mathematical Analysis itself, and also in Scattering Theory and Control Theory (cf. Chap. 18).

• Part metric

  The part metric is a metric on a domain D of $\mathbb{R}^{2}$ defined for any $x,y \in \mathbb{R}^{2}$ by

  $$\displaystyle{\sup _{f\in H^{+}}\left \vert \ln \left (\frac{f(x)} {f(y)}\right )\right \vert,}$$

  where H ⁺ is the set of all positive harmonic functions on the domain D.

  A twice-differentiable real function $f: D \rightarrow \mathbb{R}$ is called harmonic on D if its Laplacian $\Delta f = \frac{\partial ^{2}f} {\partial x_{1}^{2}} + \frac{\partial ^{2}f} {\partial x_{2}^{2}}$ vanishes on D.

• Orlicz metric

  Let M(u) be an even convex function of a real variable which is increasing for u positive, and $\lim _{u\rightarrow 0}u^{-1}M(u) =\lim _{u\rightarrow \infty }u(M(u))^{-1} = 0$. In this case the function $p(v) = M^{^{{\prime}} }(v)$ does not decrease on [0, ∞), $p(0) =\lim _{v\rightarrow 0}p(v) = 0$, and p(v) > 0 when v > 0. Writing M(u) = ∫ ₀ ^| u | p(v)dv, and defining $N(u) =\int _{ 0}^{\vert u\vert }p^{-1}(v)dv$, one obtains a pair (M(u), N(u)) of complementary functions.

  Let (M(u), N(u)) be a pair of complementary functions, and let G be a bounded closed set in $\mathbb{R}^{n}$. The Orlicz space L _M ^∗(G) is the set of Lebesgue-measurable functions f on G satisfying the following growth condition for the Orlicz norm | | f | |  _M :

  $$\displaystyle{\vert \vert f\vert \vert _{M} =\sup \left \{\int _{G}f(t)g(t)\mathit{dt}:\int _{G}N(g(t))\mathit{dt} \leq 1\right \} < \infty.}$$

  The Orlicz metric is the norm metric | | f − g | |  _M on L _M ^∗(G). It makes L _M ^∗(G) a Banach space [Orli32].

  When M(u) = u ^p, 1 < p < ∞, L _M ^∗(G) coincides with the space L _p (G), and, up to scalar factor, the L _p -norm | | f | |  _p coincides with | | f | |  _M .

  The Orlicz norm is equivalent to the Luxemburg norm $\vert \vert f\vert \vert _{(M)} =\inf \{\lambda > 0:\int _{G}M(\lambda ^{-1}f(t))\mathit{dt} \leq 1\}$; in fact, | | f | | _(M) ≤ | | f | |  _M  ≤ 2 | | f | | _(M).

• Orlicz–Lorentz metric

  Let w: (0, ∞) → (0, ∞) be a nonincreasing function. Let M: [0, ∞) → [0, ∞) be a nondecreasing and convex function with M(0) = 0. Let G be a bounded closed set in $\mathbb{R}^{n}$.

  The Orlicz–Lorentz space L _w, M (G) is the set of all Lebesgue-measurable functions f on G satisfying the following growth condition for the Orlicz–Lorentz norm | | f | |  _w, M :

  $$\displaystyle{\vert \vert f\vert \vert _{w,M} =\inf \left \{\lambda > 0:\int _{ 0}^{\infty }w(x)M\left (\frac{f^{{\ast}}(x)} {\lambda } \right )\mathit{dx} \leq 1\right \} < \infty,}$$

  where f ^∗(x) = sup{t: μ( | f | ≥ t) ≥ x} is the nonincreasing rearrangement of f.

  The Orlicz–Lorentz metric is the norm metric | | f − g | |  _w, M on L _w, M (G). It makes L _w, M (G) a Banach space.

  The Orlicz–Lorentz space is a generalization of the Orlicz space L _M ^∗(G) (cf. Orlicz metric), and the Lorentz space L _w, q (G), 1 ≤ q < ∞, of all Lebesgue-measurable functions f on G satisfying the following growth condition for the Lorentz norm:

  $$\displaystyle{\vert \vert f\vert \vert _{w,q} = \left (\int _{0}^{\infty }w(x)(f^{{\ast}}(x))^{q}\right )^{\frac{1} {q} } < \infty.}$$

• Hölder metric

  Let L ^α(G) be the set of all bounded continuous functions f defined on a subset G of $\mathbb{R}^{n}$, and satisfying the Hölder condition on G. Here, a function f satisfies the Hölder condition at a point y ∈ G with index (or order) α, 0 < α ≤ 1, and with coefficient A(y), if $\vert f(x) - f(y)\vert \leq A(y)\vert x - y\vert ^{\alpha }$ for all x ∈ G sufficiently close to y.

  If A = sup _y ∈ G (A(y)) < ∞, the Hölder condition is called uniform on G, and A is called the Hölder coefficient of G. The quantity $\vert f\vert _{\alpha } =\sup _{x,y\in G}\frac{\vert f(x)-f(y)\vert } {\vert x-y\vert ^{\alpha }}$, 0 ≤ α ≤ 1, is called the Hölder α-seminorm of f, and the Hölder norm of f is defined by

  $$\displaystyle{\vert \vert f\vert \vert _{L^{\alpha }(G)} =\sup _{x\in G}\vert f(x)\vert + \vert f\vert _{\alpha }.}$$

  The Hölder metric is the norm metric $\vert \vert f - g\vert \vert _{L^{\alpha }(G)}$ on L ^α(G). It makes L ^α(G) a Banach space.

• Sobolev metric

  The Sobolev space W ^k, p is a subset of an L _p -space such that f and its derivatives up to order k have a finite L _p -norm. Formally, given a subset G of $\mathbb{R}^{n}$, define

  $$\displaystyle{W^{k,p} = W^{k,p}(G) =\{ f \in L_{ p}(G): f^{(i)} \in L_{ p}(G),1 \leq i \leq k\},}$$

  where $f^{(i)} = \partial _{x_{1}}^{\alpha _{1}}\ldots \partial _{ x_{n}}^{\alpha _{n}}f$, $\alpha _{1} +\ldots +\alpha _{n} = i$, and the derivatives are taken in a weak sense. The Sobolev norm on W ^k, p is defined by

  $$\displaystyle{\vert \vert f\vert \vert _{k,p} =\sum _{ i=0}^{k}\vert \vert f^{(i)}\vert \vert _{ p}.}$$

  In fact, it is enough to take only the first and last in the sequence, i.e., the norm defined by $\vert \vert f\vert \vert _{k,p} = \vert \vert f\vert \vert _{p} + \vert \vert f^{(k)}\vert \vert _{p}$ is equivalent to the norm above.

  For p = ∞, the Sobolev norm is equal to the essential supremum of | f | : $\vert \vert f\vert \vert _{k,\infty } = \mathit{ess}\sup _{x\in G}\vert f(x)\vert$, i.e., it is the infimum of all numbers $a \in \mathbb{R}$ for which | f(x) |  > a on a set of measure zero.

  The Sobolev metric is the norm metric | | f − g | |  _k, p on W ^k, p. It makes W ^k, p a Banach space.

  The Sobolev space W ^k, 2 is denoted by H ^k. It is a Hilbert space for the inner product $\langle f,g\rangle _{k} =\sum _{ i=1}^{k}\langle f^{(i)},g^{(i)}\rangle _{L_{2}} =\sum _{ i=1}^{k}\int _{G}f^{(i)}\overline{g}^{(i)}\mu (d\omega )$.

• Variable exponent space metrics

  Let G be a nonempty open subset of $\mathbb{R}^{n}$, and let $p: G \rightarrow [1,\infty )$ be a measurable bounded function, called a variable exponent. The variable exponent Lebesgue space L _p(. )(G) is the set of all measurable functions $f: G \rightarrow \mathbb{R}$ for which the modular $\varrho _{p(.)}(f) =\int _{G}\vert f(x)\vert ^{p(x)}\mathit{dx}$ is finite. The Luxemburg norm on this space is defined by

  $$\displaystyle{\vert \vert f\vert \vert _{p(.)} =\inf \{\lambda > 0:\varrho _{p(.)}(f/\lambda ) \leq 1\}.}$$

  The variable exponent Lebesgue space metric is the norm metric | | f − g | |  _p(. ) on L _p(. )(G).

  A variable exponent Sobolev space W ^1, p(. )(G) is a subspace of L _p(. )(G) consisting of functions f whose distributional gradient exists almost everywhere and satisfies the condition | ∇f | ∈ L _p(. )(G). The norm

  $$\displaystyle{\vert \vert f\vert \vert _{1,p(.)} = \vert \vert f\vert \vert _{p(.)} + \vert \vert \nabla f\vert \vert _{p(.)}}$$

  makes W ^1, p(. )(G) a Banach space. The variable exponent Sobolev space metric is the norm metric | | f − g | | _1, p(. ) on W ^1, p(. ).

• Schwartz metric

  The Schwartz space (or space of rapidly decreasing functions) $S(\mathbb{R}^{n})$ is the class of all Schwartz functions, i.e., infinitely-differentiable functions $f: \mathbb{R}^{n} \rightarrow \mathbb{C}$ that decrease at infinity, as do all their derivatives, faster than any inverse power of x. More precisely, f is a Schwartz function if we have the following growth condition:

  $$\displaystyle{\vert \vert f\vert \vert _{\alpha \beta } =\sup _{x\in \mathbb{R}^{n}}\left \vert x_{1}^{\beta _{1} }\ldots x_{n}^{\beta _{n} }\frac{\partial ^{\alpha _{1}+\ldots +\alpha _{n}}f(x_{1},\ldots,x_{n})} {\partial x_{1}^{\alpha _{1}}\ldots \partial x_{n}^{\alpha _{n}}} \right \vert < \infty }$$

  for any nonnegative integer vectors α and β. The family of seminorms | | . | |  _{α β} defines a locally convex topology of $S(\mathbb{R}^{n})$ which is metrizable and complete. The Schwartz metric is a metric on $S(\mathbb{R}^{n})$ which can be obtained using this topology (cf. countably normed space in Chap. 2).

  The corresponding metric space on $S(\mathbb{R}^{n})$ is a Fréchet space in the sense of Functional Analysis, i.e., a locally convex F-space.

• Bregman quasi-distance

  Let $G \subset \mathbb{R}^{n}$ be a closed set with the nonempty interior G ⁰. Let f be a Bregman function with zone G.

  The Bregman quasi-distance $D_{f}: G \times G^{0} \rightarrow \mathbb{R}_{\geq 0}$ is defined by

  $$\displaystyle{D_{f}(x,y) = f(x) - f(y) -\langle \nabla f(y),x - y\rangle,}$$

  where $\nabla f = ( \frac{\partial f} {\partial x_{1}},\ldots \frac{\partial f} {\partial x_{n}})$. D _f (x, y) = 0 if and only if x = y. Also $D_{f}(x,y) + D_{f}(y,z) - D_{f}(x,z) =\langle \nabla f(z) -\nabla f(y),x - y\rangle$ but, in general, D _f does not satisfy the triangle inequality, and is not symmetric.

  A real-valued function f whose effective domain contains G is called a Bregman function with zone G if the following conditions hold:

  • 1. f is continuously differentiable on G ⁰;

  • 2. f is strictly convex and continuous on G;

  • 3. For all $\delta \in \mathbb{R}$ the partial level sets $\Gamma (x,\delta ) =\{ y \in G^{0}: D_{f}(x,y) \leq \delta \}$ are bounded for all x ∈ G;

  • 4. If {y _n } _n  ⊂ G ⁰ converges to y ^∗, then D _f (y ^∗, y _n ) converges to 0;

  • 5. If {x _n } _n  ⊂ G and {y _n } _n  ⊂ G ⁰ are sequences such that {x _n } _n is bounded, lim _{n → ∞} y _n  = y ^∗, and $\lim _{n\rightarrow \infty }D_{f}(x_{n},y_{n}) = 0$, then $\lim _{n\rightarrow \infty }x_{n} = y^{{\ast}}$.

  When $G = \mathbb{R}^{n}$, a sufficient condition for a strictly convex function to be a Bregman function has the form: $\lim _{\vert \vert x\vert \vert \rightarrow \infty }\frac{f(x)} {\vert \vert x\vert \vert } = \infty$.

2 Metrics on Linear Operators

A linear operator is a function T: V → W between two vector spaces V, W over a field $\mathbb{F}$, that is compatible with their linear structures, i.e., for any x, y ∈ V and any scalar $k \in \mathbb{F}$, we have the following properties: $T(x + y) = T(x) + T(y)$, and T(kx) = kT(x).

• Operator norm metric

  Consider the set of all linear operators from a normed space (V, | | . | |  _V ) into a normed space (W, | | . | |  _W ). The operator norm | | T | | of a linear operator T: V → W is defined as the largest value by which T stretches an element of V, i.e.,

  $$\displaystyle{\vert \vert T\vert \vert =\sup _{\vert \vert v\vert \vert _{V }\neq 0}\frac{\vert \vert T(v)\vert \vert _{W}} {\vert \vert v\vert \vert _{V }} =\sup _{\vert \vert v\vert \vert _{V }=1}\vert \vert T(v)\vert \vert _{W} =\sup _{\vert \vert v\vert \vert _{V }\leq 1}\vert \vert T(v)\vert \vert _{W}.}$$

  A linear operator T: V → W from a normed space V into a normed space W is called bounded if its operator norm is finite. For normed spaces, a linear operator is bounded if and only if it is continuous.

  The operator norm metric is a norm metric on the set B(V, W) of all bounded linear operators from V into W, defined by

  $$\displaystyle{\vert \vert T - P\vert \vert.}$$

  The space (B(V, W), | | . | | ) is called the space of bounded linear operators. This metric space is complete if W is. If V = W is complete, the space B(V, V ) is a Banach algebra, as the operator norm is a submultiplicative norm.

  A linear operator T: V → W from a Banach space V into another Banach space W is called compact if the image of any bounded subset of V is a relatively compact subset of W. Any compact operator is bounded (and, hence, continuous). The space (K(V, W), | | . | | ) on the set K(V, W) of all compact operators from V into W with the operator norm | | . | | is called the space of compact operators.

• Nuclear norm metric

  Let B(V, W) be the space of all bounded linear operators mapping a Banach space (V, | | . | |  _V ) into another Banach space (W, | | . | |  _W ). Let the Banach dual of V be denoted by $V ^{^{{\prime}} }$, and the value of a functional $x^{^{{\prime}} } \in V ^{^{{\prime}} }$ at a vector x ∈ V by $\langle x,x^{^{{\prime}} }\rangle$.

  A linear operator T ∈ B(V, W) is called a nuclear operator if it can be represented in the form $x\mapsto T(x) =\sum _{ i=1}^{\infty }\langle x,x_{i}^{^{{\prime}} }\rangle y_{i}$, where $\{x_{i}^{^{{\prime}} }\}_{i}$ and {y _i } _i are sequences in $V ^{^{{\prime}} }$ and W, respectively, such that $\sum _{i=1}^{\infty }\vert \vert x_{i}^{^{{\prime}} }\vert \vert _{V ^{^{{\prime}}}}\vert \vert y_{i}\vert \vert _{W} < \infty$. This representation is called nuclear, and can be regarded as an expansion of T as a sum of operators of rank 1 (i.e., with one-dimensional range). The nuclear norm of T is defined as

  $$\displaystyle{\vert \vert T\vert \vert _{\mathit{nuc}} =\inf \sum _{ i=1}^{\infty }\vert \vert x_{ i}^{^{{\prime}} }\vert \vert _{V ^{^{{\prime}}}}\vert \vert y_{i}\vert \vert _{W},}$$

  where the infimum is taken over all possible nuclear representations of T.

  The nuclear norm metric is the norm metric | | T − P | |  _nuc on the set N(V, W) of all nuclear operators mapping V into W. The space (N(V, W), | | . | |  _nuc ), called the space of nuclear operators, is a Banach space.

  A nuclear space is defined as a locally convex space for which all continuous linear functions into an arbitrary Banach space are nuclear operators. A nuclear space is constructed as a projective limit of Hilbert spaces H _α with the property that, for each α ∈ I, one can find β ∈ I such that H _β  ⊂ H _α , and the embedding operator H _β  ∋ x → x ∈ H _α is a Hilbert–Schmidt operator. A normed space is nuclear if and only if it is finite-dimensional.

• Finite nuclear norm metric

  Let F(V, W) be the space of all linear operators of finite rank (i.e., with finite-dimensional range) mapping a Banach space (V, | | . | |  _V ) into another Banach space (W, | | . | |  _W ). A linear operator T ∈ F(V, W) can be represented in the form $x\mapsto T(x) =\sum _{ i=1}^{n}\langle x,x_{i}^{^{{\prime}} }\rangle y_{i}$, where $\{x_{i}^{^{{\prime}} }\}_{i}$ and {y _i } _i are sequences in $V ^{^{{\prime}} }$ (Banach dual of V ) and W, respectively, and $\langle x,x^{^{{\prime}} }\rangle$ is the value of a functional $x^{^{{\prime}} } \in V ^{^{{\prime}} }$ at a vector x ∈ V. The finite nuclear norm of T is defined as

  $$\displaystyle{\vert \vert T\vert \vert _{\mathit{fnuc}} =\inf \sum _{ i=1}^{n}\vert \vert x_{ i}^{^{{\prime}} }\vert \vert _{V ^{^{{\prime}}}}\vert \vert y_{i}\vert \vert _{W},}$$

  where the infimum is taken over all possible finite representations of T.

  The finite nuclear norm metric is the norm metric | | T − P | |  _fnuc on F(V, W). The space (F(V, W), | | . | |  _fnuc ) is called the space of operators of finite rank. It is a dense linear subspace of the space of nuclear operators N(V, W).

• Hilbert–Schmidt norm metric

  Consider the set of all linear operators from a Hilbert space $(H_{1},\vert \vert.\vert \vert _{H_{1}})$ into a Hilbert space $(H_{2},\vert \vert.\vert \vert _{H_{2}})$. The Hilbert–Schmidt norm | | T | |  _HS of a linear operator T: H ₁ → H ₂ is defined by

  $$\displaystyle{\vert \vert T\vert \vert _{\mathit{HS}} = \left (\sum _{\alpha \in I}\vert \vert T(e_{\alpha })\vert \vert _{H_{2}}^{2}\right )^{1/2},}$$

  where (e _α ) _α ∈ I is an orthonormal basis in H ₁. A linear operator T: H ₁ → H ₂ is called a Hilbert–Schmidt operator if | | T | |  _HS ² < ∞.

  The Hilbert–Schmidt norm metric is the norm metric | | T − P | |  _HS on the set S(H ₁, H ₂) of all Hilbert–Schmidt operators from H ₁ into H ₂. In Euclidean space | | . | |  _HS is also called Frobenius norm; cf. Frobenius norm metric in Chap. 12.

  For $H_{1} = H_{2} = H$, the algebra S(H, H) = S(H) with the Hilbert–Schmidt norm is a Banach algebra. It contains operators of finite rank as a dense subset, and is contained in the space K(H) of compact operators. An inner product $\langle,\rangle _{\mathit{HS}}$ on S(H) is defined by $\langle T,P\rangle _{\mathit{HS}} =\sum _{\alpha \in I}\langle T(e_{\alpha }),P(e_{\alpha })\rangle$, and $\vert \vert T\vert \vert _{\mathit{HS}} =\langle T,T\rangle _{\mathit{HS}}^{1/2}$. So, S(H) is a Hilbert space (independent of the chosen basis (e _α ) _α ∈ I ).

• Trace-class norm metric

  Given a Hilbert space H, the trace-class norm of a linear operator T: H → H is

  $$\displaystyle{\vert \vert T\vert \vert _{\mathit{tc}} =\sum _{\alpha \in I}\langle \vert T\vert (e_{\alpha }),e_{\alpha }\rangle,}$$

  where | T | is the absolute value of T in the Banach algebra B(H) of all bounded operators from H into itself, and (e _α ) _α ∈ I is an orthonormal basis of H.

  An operator T: H → H is called a trace-class operator if | | T | |  _tc  < ∞. Any such operator is the product of two Hilbert–Schmidt operators.

  The trace-class norm metric is the norm metric | | T − P | |  _tc on the set L(H) of all trace-class operators from H into itself.

  The set L(H) with the norm | | . | |  _tc forms a Banach algebra which is contained in the algebra K(H) (of all compact operators from H into itself), and contains the algebra S(H) of all Hilbert–Schmidt operators from H into itself.

• Schatten p -class norm metric

  Let 1 ≤ p < ∞. Given a separable Hilbert space H, the Schatten p-class norm of a compact linear operator T: H → H is defined by

  $$\displaystyle{\vert \vert T\vert \vert _{\mathit{Sch}}^{p} = \left (\sum _{ n}\vert s_{n}\vert ^{p}\right )^{\frac{1} {p} },}$$

  where {s _n } _n is the sequence of singular values of T. A compact operator T: H → H is called a Schatten p-class operator if | | T | |  _Sch ^p < ∞.

  The Schatten p -class norm metric is the norm metric | | T − P | |  _Sch ^p on the set S _p (H) of all Schatten p-class operators from H onto itself. The set S _p (H) with the norm | | . | |  _Sch ^p forms a Banach space. S ₁(H) is the trace-class of H, and S ₂(H) is the Hilbert–Schmidt class of H. Cf. Schatten norm metric (in Chap. 12) for which trace and Frobenius norm metrics are cases p = 1 and p = 2, respectively.

• Continuous dual space

  For any vector space V over some field, its algebraic dual space is the set of all linear functionals on V.

  Let (V, | | . | | ) be a normed vector space. Let $V ^{^{{\prime}} }$ be the set of all continuous linear functionals T from V into the base field ($\mathbb{R}$ or $\mathbb{C}$). Let $\vert \vert.\vert \vert ^{^{{\prime}} }$ be the operator norm on $V ^{^{{\prime}} }$ defined by

  $$\displaystyle{\vert \vert T\vert \vert ^{^{{\prime}} } =\sup _{\vert \vert x\vert \vert \leq 1}\vert T(x)\vert.}$$

  The space $(V ^{^{{\prime}} },\vert \vert.\vert \vert ^{^{{\prime}} })$ is a Banach space which is called the continuous dual (or Banach dual) of (V, | | . | | ).

  The continuous dual of the metric space l _p ⁿ (l _p ^∞) is l _q ⁿ (l _q ^∞, respectively), where q is defined by $\frac{1} {p} + \frac{1} {q} = 1$. The continuous dual of l ₁ ⁿ (l ₁ ^∞) is l _∞ ⁿ (l _∞ ^∞, respectively).

• Distance constant of operator algebra

  Let $\mathcal{A}$ be an subalgebra of B(H), the algebra of all bounded operators on a Hilbert space H. For any operator T ∈ B(H), let P be a projection, P ^⊥ be its orthogonal complement and $\beta (T,\mathcal{A}) =\sup \{ \vert \vert P^{\perp }TP\vert \vert: P^{\perp }\mathcal{A}P = (0)\}$.

  Let $\mathit{dist}(T,\mathcal{A}) =\inf _{A\in \mathcal{A}}\vert \vert T - A\vert \vert$ be the distance of T to algebra $\mathcal{A}$; cf. matrix nearness problems in Chap. 12. It holds $\mathit{dist}(T,\mathcal{A}) \geq \beta (T,\mathcal{A})$.

  The algebra $\mathcal{A}$ is reflexive if $\beta (T,\mathcal{A}) = 0$ implies $T \in \mathcal{A}$; it is hyperreflexive if there exists a constant C ≥ 1 such that, for any operator T ∈ B(H), it holds

  $$\displaystyle{\mathit{dist}(T,\mathcal{A}) \leq C\beta (T,\mathcal{A}).}$$

  The smallest such C is called the distance constant of the algebra $\mathcal{A}$.

  In the case of a reflexive algebra of matrices with nonzero entries specified by a given pattern, the problem of finding the distance constant can be formulated as a matrix-filling problem: given a partially completed matrix, fill in the remaining entries so that the operator norm of the resulting complete matrix is minimized.