Metrics on Normed Structures

Abstract

In this chapter we consider a special class of metrics defined on some normed structures, as the norm of the difference between two given elements. This structure can be a group (with a group norm), a vector space (with a vector norm or, simply, a norm), a vector lattice (with a Riesz norm), a field (with a valuation), etc.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this chapter we consider a special class of metrics defined on some normed structures, as the norm of the difference between two given elements. This structure can be a group (with a group norm), a vector space (with a vector norm or, simply, a norm), a vector lattice (with a Riesz norm), a field (with a valuation), etc.

Any norm is subadditive, i.e., triangle inequality $\vert \vert x + y\vert \vert \leq \vert \vert x\vert \vert + \vert \vert y\vert \vert$ holds. A norm is submultiplicative if multiplicative triangle inequality  | | xy | | ≤ | | x | | | | y | | holds.

• Group norm metric

  A group norm metric is a metric on a group (G, +, 0) defined by

  $$\displaystyle{\vert \vert x + (-y)\vert \vert = \vert \vert x - y\vert \vert,}$$

  where | | . | | is a group norm on G, i.e., a function $\vert \vert.\vert \vert: G \rightarrow \mathbb{R}$ such that, for all x, y ∈ G, we have the following properties:

  • 1.  | | x | | ≥ 0, with | | x | | = 0 if and only if x = 0;

  • 2. $\vert \vert x\vert \vert = \vert \vert - x\vert \vert$;

  • 3. $\vert \vert x + y\vert \vert \leq \vert \vert x\vert \vert + \vert \vert y\vert \vert$ (triangle inequality).

  Any group norm metric d is right-invariant, i.e., $d(x,y) = d(x + z,y + z)$ for any x, y, z ∈ G. Conversely, any right-invariant (as well as any left-invariant, and, in particular, any bi-invariant) metric d on G is a group norm metric, since one can define a group norm on G by | | x | | = d(x, 0).

• F -norm metric

  A vector space (or linear space) over a field $\mathbb{F}$ is a set V equipped with operations of vector addition $+: V \times V \rightarrow V$ and scalar multiplication $\cdot: \mathbb{F} \times V \rightarrow V$ such that (V, +, 0) forms an Abelian group (where 0 ∈ V is the zero vector), and, for all vectors x, y ∈ V and any scalars $a,b \in \mathbb{F}$, we have the following properties: 1 ⋅ x = x (where 1 is the multiplicative unit of $\mathbb{F}$), (ab) ⋅ x = a ⋅ (b ⋅ x), $(a + b) \cdot x = a \cdot x + b \cdot x$, and $a \cdot (x + y) = a \cdot x + a \cdot y$.

  A vector space over the field $\mathbb{R}$ of real numbers is called a real vector space. A vector space over the field $\mathbb{C}$ of complex numbers is called complex vector space.

  A F-norm metric is a metric on a real (complex) vector space V defined by

  $$\displaystyle{\vert \vert x - y\vert \vert _{F},}$$

  where | | . | |  _F is an F-norm on V, i.e., a function $\vert \vert.\vert \vert _{F}: V \rightarrow \mathbb{R}$ such that, for all x, y ∈ V and for any scalar a with | a | = 1, we have the following properties:

  • 1.  | | x | |  _F  ≥ 0, with | | x | |  _F  = 0 if and only if x = 0;

  • 2. $\vert \vert \mathit{ax}\vert \vert _{F} \leq \vert \vert x\vert \vert _{F}$ if | a | ≤ 1;

  • 3. $\lim _{a\rightarrow 0}\vert \vert \mathit{ax}\vert \vert _{F} = 0$;

  • 4. $\vert \vert x + y\vert \vert _{F} \leq \vert \vert x\vert \vert _{F} + \vert \vert y\vert \vert _{F}$ (triangle inequality).

  An F-norm is called p-homogeneous if $\vert \vert \mathit{ax}\vert \vert _{F} = \vert a\vert ^{p}\vert \vert x\vert \vert _{F}$ for any scalar a.

  Any F-norm metric d is a translation invariant metric , i.e., $d(x,y) = d(x + z,y + z)$ for all x, y, z ∈ V. Conversely, if d is a translation invariant metric on V, then | | x | |  _F  = d(x, 0) is an F-norm on V.

• F ^∗ -metric

  An F ^∗ -metric is an F-norm metric | | x − y | |  _F on a real (complex) vector space V such that the operations of scalar multiplication and vector addition are continuous with respect to | | . | |  _F . Thus | | . | |  _F is a function $\vert \vert.\vert \vert _{F}: V \rightarrow \mathbb{R}$ such that, for all x, y, x _n  ∈ V and for all scalars a, a _n , we have the following properties:

  • 1.  | | x | |  _F  ≥ 0, with | | x | |  _F  = 0 if and only if x = 0;

  • 2. $\vert \vert \mathit{ax}\vert \vert _{F} = \vert \vert x\vert \vert _{F}$ for all a with | a | = 1;

  • 3. $\vert \vert x + y\vert \vert _{F} \leq \vert \vert x\vert \vert _{F} + \vert \vert y\vert \vert _{F}$;

  • 4. $\vert \vert a_{n}x\vert \vert _{F} \rightarrow 0$ if a _n  → 0;

  • 5. $\vert \vert \mathit{ax}_{n}\vert \vert _{F} \rightarrow 0$ if x _n  → 0;

  • 6. $\vert \vert a_{n}x_{n}\vert \vert _{F} \rightarrow 0$ if a _n  → 0, x _n  → 0.

  The metric space (V, | | x − y | |  _F ) with an F ^∗-metric is called a nF ^∗ -space . Equivalently, an F ^∗-space is a metric space (V, d) with a translation invariant metric d such that the operation of scalar multiplication and vector addition are continuous with respect to this metric.

  A complete F ^∗-space is called an F -space . A locally convex F-space is known as a Fréchet space (cf. Chap. 2) in Functional Analysis.

  A modular space is an F ^∗-space (V, | | . | |  _F ) in which the F-norm | | . | |  _F is defined by

  $$\displaystyle{\vert \vert x\vert \vert _{F} =\inf \{\lambda> 0:\rho \left (\frac{x} {\lambda } \right ) <\lambda \},}$$

  and ρ is a metrizing modular on V, i.e., a function ρ: V → [0, ∞] such that, for all x, y, x _n  ∈ V and for all scalars a, a _n , we have the following properties:

  • 1. ρ(x) = 0 if and only if x = 0;

  • 2. ρ(ax) = ρ(x) implies | a | = 1;

  • 3. $\rho (\mathit{ax} + \mathit{by}) \leq \rho (x) +\rho (y)$ implies $a,b \geq 0,a + b = 1$;

  • 4. ρ(a _n x) → 0 if a _n  → 0 and ρ(x) < ∞;

  • 5. ρ(ax _n ) → 0 if ρ(x _n ) → 0 (metrizing property);

  • 6. For any x ∈ V, there exists k > 0 such that ρ(kx) < ∞.

• Norm metric

  A norm metric is a metric on a real (complex) vector space V defined by

  $$\displaystyle{\vert \vert x - y\vert \vert,}$$

  where | | . | | is a norm on V, i.e., a function $\vert \vert.\vert \vert: V \rightarrow \mathbb{R}$ such that, for all x, y ∈ V and for any scalar a, we have the following properties:

  • 1.  | | x | | ≥ 0, with | | x | | = 0 if and only if x = 0;

  • 2.  | | ax | | = | a | | | x | | ;

  • 3. $\vert \vert x + y\vert \vert \leq \vert \vert x\vert \vert + \vert \vert y\vert \vert$ (triangle inequality).

  Therefore, a norm | | . | | is a 1-homogeneous F-norm. The vector space (V, | | . | | ) is called a normed vector space or, simply, normed space.

  Any metric space can be embedded isometrically in some normed vector space as a closed linearly independent subset. Every finite-dimensional normed space is complete, and all norms on it are equivalent.

  In general, the norm | | . | | is equivalent (Maligranda, 2008) to the norm

  $$\displaystyle{\vert \vert x\vert \vert _{u,p} = (\vert \vert x + \vert \vert x\vert \vert \cdot u\vert \vert ^{p} + \vert \vert x -\vert \vert x\vert \vert \cdot u\vert \vert ^{p})^{\frac{1} {p} },}$$

  introduced, for any u ∈ V and p ≥ 1, by Odell and Schlumprecht, 1998.

  The norm-angular distance between x and y is defined (Clarkson, 1936) by

  $$\displaystyle{d(x,y) = \vert \vert \frac{x} {\vert \vert x\vert \vert }- \frac{y} {\vert \vert y\vert \vert }\vert \vert.}$$

  The following sharpening of the triangle inequality (Maligranda, 2003) holds:

  $$\displaystyle\begin{array}{rcl} & & \frac{\vert \vert x - y\vert \vert -\vert \vert \vert x\vert \vert -\vert \vert y\vert \vert \vert } {\min \{\vert \vert x\vert \vert,\vert \vert y\vert \vert \}} \leq d(x,y) \leq \frac{\vert \vert x - y\vert \vert + \vert \vert \vert x\vert \vert -\vert \vert y\vert \vert \vert } {\max \{\vert \vert x\vert \vert,\vert \vert y\vert \vert \}},\mbox{ i.e.,} {}\\ & & \qquad \quad (2 - d(x,-y))\min \{\vert \vert x\vert \vert,\vert \vert y\vert \vert \}\leq \vert \vert x\vert \vert + \vert \vert y\vert \vert -\vert \vert x + y\vert \vert {}\\ &&\qquad \qquad \qquad \quad \leq (2 - d(x,-y))\max \{\vert \vert x\vert \vert,\vert \vert y\vert \vert \}. {}\\ \end{array}$$

  Dragomir, 2004, call $\vert \int _{a}^{b}f(x)\mathit{dx}\vert \leq \int _{a}^{b}\vert f(x)\vert \mathit{dx}$ continuous triangle inequality.

• Reverse triangle inequality

  The triangle inequality $\vert \vert x + y\vert \vert \leq \vert \vert x\vert \vert + \vert \vert y\vert \vert$ in a normed space (V, | | . | | ) is equivalent to the following inequality, for any $x_{1},\ldots,x_{n} \in V$ with n ≥ 2:

  $$\displaystyle{\vert \vert \sum _{i=1}^{n}x_{ i}\vert \vert \leq \sum _{i=1}^{n}\vert \vert x_{ i}\vert \vert.}$$

  If in the normed space (V, | | . | | ), for some C ≥ 1 one has

  $$\displaystyle{C\vert \vert \sum _{i=1}^{n}x_{ i}\vert \vert \geq \sum _{i=1}^{n}\vert \vert x_{ i}\vert \vert,}$$

  then this inequality is called the reverse triangle inequality.

  This term is used, sometimes, also for the inverse triangle inequality (cf. kinematic metric in Chap. 26) and for the eventual inequality Cd(x, z) ≥ d(x, y) + d(y, z) with C ≥ 1 in a metric space (X, d).

  The triangle inequality $\vert \vert x + y\vert \vert \leq \vert \vert x\vert \vert + \vert \vert y\vert \vert$, for any x, y ∈ V, in a normed space (V, | | . | | ) is, for any number q > 1, equivalent (Belbachir, Mirzavaziri and Moslenian, 2005) to the following inequality:

  $$\displaystyle{\vert \vert x + y\vert \vert ^{q} \leq 2^{q-1}(\vert \vert x\vert \vert ^{q} + \vert \vert y\vert \vert ^{q}).}$$

  The parallelogram inequality $\vert \vert x + y\vert \vert ^{2} \leq 2(\vert \vert x\vert \vert ^{2} + \vert \vert y\vert \vert ^{2})$ is the case q = 2 of above.

  Given a number q, 0 < q ≤ 1, the norm is called q-subadditive if $\vert \vert x + y\vert \vert ^{q} \leq \vert \vert x\vert \vert ^{q} + \vert \vert y\vert \vert ^{q}$ holds for x, y ∈ V.

• Seminorm semimetric

  A seminorm semimetric on a real (complex) vector space V is defined by

  $$\displaystyle{\vert \vert x - y\vert \vert,}$$

  where | | . | | is a seminorm (or pseudo-norm) on V, i.e., a function $\vert \vert.\vert \vert: V \rightarrow \mathbb{R}$ such that, for all x, y ∈ V and for any scalar a, we have the following properties:

  • 1.  | | x | | ≥ 0, with | | 0 | | = 0;

  • 2.  | | ax | | = | a | | | x | | ;

  • 3. $\vert \vert x + y\vert \vert \leq \vert \vert x\vert \vert + \vert \vert y\vert \vert$ (triangle inequality).

  The vector space (V, | | . | | ) is called a seminormed vector space. Many normed vector spaces, in particular, Banach spaces, are defined as the quotient space by the subspace of elements of seminorm zero.

  A quasi-normed space is a vector space V, on which a quasi-norm is given. A quasi-norm on V is a nonnegative function $\vert \vert.\vert \vert: V \rightarrow \mathbb{R}$ which satisfies the same axioms as a norm, except for the triangle inequality which is replaced by the weaker requirement: there exists a constant C > 0 such that, for all x, y ∈ V, the following C-triangle inequality (cf. near-metric in Chap. 1) holds:

  $$\displaystyle{\vert \vert x + y\vert \vert \leq C(\vert \vert x\vert \vert + \vert \vert y\vert \vert )}$$

  An example of a quasi-normed space, that is not normed, is the Lebesgue space $L_{p}(\Omega )$ with 0 < p < 1 in which a quasi-norm is defined by

  $$\displaystyle{\vert \vert f\vert \vert = \left (\int _{\Omega }\vert f(x)\vert ^{p}\mathit{dx}\right )^{1/p},f \in L_{ p}(\Omega ).}$$

• Banach space

  A Banach space (or B-space) is a complete metric space (V, | | x − y | | ) on a vector space V with a norm metric | | x − y | | . Equivalently, it is the complete normed space (V, | | . | | ). In this case, the norm | | . | | on V is called the Banach norm. Some examples of Banach spaces are:

  • 1. l _p ⁿ -spaces, l _p ^∞ -spaces, 1 ≤ p ≤ ∞, $n \in \mathbb{N}$;

  • 2. The space C of convergent numerical sequences with the norm $\vert \vert x\vert \vert =\sup _{n}\vert x_{n}\vert$;

  • 3. The space C ₀ of numerical sequences which converge to zero with the norm $\vert \vert x\vert \vert =\max _{n}\vert x_{n}\vert$;

  • 4. The space C _[a, b] ^p, 1 ≤ p ≤ ∞, of continuous functions on [a, b] with the L _p -norm $\vert \vert f\vert \vert _{p} = (\int _{a}^{b}\vert f(t)\vert ^{p}\mathit{dt})^{\frac{1} {p} }$;

  • 5. The space C _K of continuous functions on a compactum K with the norm | | f | | = max _t ∈ K  | f(t) | ;

  • 6. The space (C _[a, b])ⁿ of functions on [a, b] with continuous derivatives up to and including the order n with the norm $\vert \vert f\vert \vert _{n} =\sum _{ k=0}^{n}\max _{a\leq t\leq b}\vert f^{(k)}(t)\vert$;

  • 7. The space $C^{n}[I^{m}]$ of all functions defined in an m-dimensional cube that are continuously differentiable up to and including the order n with the norm of uniform boundedness in all derivatives of order at most n;

  • 8. The space M _[a, b] of bounded measurable functions on [a, b] with the norm

    $$\displaystyle{\vert \vert f\vert \vert = \mathit{ess}\sup _{a\leq t\leq b}\vert f(t)\vert =\inf _{e,\mu (e)=0}\sup _{t\in [a,b]\setminus e}\vert f(t)\vert;}$$

  • 9. The space $A(\Delta )$ of functions analytic in the open unit disk $\Delta =\{ z \in \mathbb{C}: \vert z\vert <1\}$ and continuous in the closed disk $\overline{\Delta }$ with the norm $\vert \vert f\vert \vert =\max _{z\in \overline{\Delta }}\vert f(z)\vert$;

  • 10. The Lebesgue spaces $L_{p}(\Omega )$, 1 ≤ p ≤ ∞;

  • 11. The Sobolev spaces $W^{k,p}(\Omega )$, $\Omega \subset \mathbb{R}^{n}$, $1 \leq p \leq \infty$, of functions f on $\Omega$ such that f and its derivatives, up to some order k, have a finite L _p -norm, with the norm $\vert \vert f\vert \vert _{k,p} =\sum _{ i=0}^{k}\vert \vert f^{(i)}\vert \vert _{p}$;

  • 12. The Bohr space AP of almost periodic functions with the norm

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

  A finite-dimensional real Banach space is called a Minkowskian space. A norm metric of a Minkowskian space is called a Minkowskian metric (cf. Chap. 6). In particular, any l _p -metric is a Minkowskian metric.

  All n-dimensional Banach spaces are pairwise isomorphic; the set of such spaces becomes compact if one introduces the Banach–Mazur distance by $d_{\mathit{BM}}(V,W) =\ln \inf _{T}\vert \vert T\vert \vert \cdot \vert \vert T^{-1}\vert \vert$, where the infimum is taken over all operators which realize an isomorphism $T: V \rightarrow W$.

• l _p -metric

  The l _p -metric $d_{l_{p}}$, 1 ≤ p ≤ ∞, is a norm metric on $\mathbb{R}^{n}$ (or on $\mathbb{C}^{n}$), defined by

  $$\displaystyle{\vert \vert x - y\vert \vert _{p},}$$

  where the l _p -norm | | . | |  _p is defined by

  $$\displaystyle{\vert \vert x\vert \vert _{p} = (\sum _{i=1}^{n}\vert x_{ i}\vert ^{p})^{\frac{1} {p} }.}$$

  For p = ∞, we obtain $\vert \vert x\vert \vert _{\infty } =\lim _{p\rightarrow \infty }\root{p}\of{\sum _{i=1}^{n}\vert x_{i}\vert ^{p}} =\max _{1\leq i\leq n}\vert x_{i}\vert$. The metric space $(\mathbb{R}^{n},d_{l_{p}})$ is abbreviated as $l_{p}^{n}$ and is called l _p ⁿ -space.

  The l _p -metric, 1 ≤ p ≤ ∞, on the set of all sequences $x =\{ x_{n}\}_{n=1}^{\infty }$ of real (complex) numbers, for which the sum $\sum _{i=1}^{\infty }\vert x_{i}\vert ^{p}$ (for p = ∞, the sum $\sum _{i=1}^{\infty }\vert x_{i}\vert$) is finite, is

  $$\displaystyle{(\sum _{i=1}^{\infty }\vert x_{ i} - y_{i}\vert ^{p})^{\frac{1} {p} }.}$$

  For p = ∞, we obtain $\max _{i\geq 1}\vert x_{i} - y_{i}\vert$. This metric space is abbreviated as $l_{p}^{\infty }$ and is called l _p ^∞ -space.

  Most important are l ₁-, l ₂- and l _∞ -metrics. Among l _p -metrics, only l ₁- and l _∞ -metrics are crystalline metrics , i.e., metrics having polygonal unit balls. On $\mathbb{R}$ all l _p -metrics coincide with the natural metric (cf. Chap. 12) | x − y | .

  The l ₂-norm $\vert \vert (x_{1},x_{2})\vert \vert _{2} = \sqrt{x_{1 }^{2 } + x_{2 }^{2}}$ on $\mathbb{R}^{2}$ is also called Pythagorean addition of the numbers x ₁ and x ₂. Under this commutative operation, $\mathbb{R}$ form a semigroup, and $\mathbb{R}_{\geq 0}$ form a monoid (semigroup with identity, 0).

• Euclidean metric

  The Euclidean metric (or Pythagorean distance , as-the-crow-flies distance , beeline distance ) d _E is the metric on $\mathbb{R}^{n}$ defined by

  $$\displaystyle{\vert \vert x - y\vert \vert _{2} = \sqrt{(x_{1 } - y_{1 } )^{2 } +\ldots +(x_{n } - y_{n } )^{2}}.}$$

  It is the ordinary l ₂ -metric on $\mathbb{R}^{n}$. The metric space $(\mathbb{R}^{n},d_{E})$ is abbreviated as $\mathbb{E}^{n}$ and is called Euclidean space “Euclidean space” stands for the case n = 3, as opposed, for n = 2, to Euclidean plane and, for n = 1, Euclidean (or real) line.

  In fact, $\mathbb{E}^{n}$ is an inner product space (and even a Hilbert space), i.e., $d_{E}(x,y) = \vert \vert x - y\vert \vert _{2} = \sqrt{\langle x - y, x - y\rangle }$, where $\langle x,y\rangle$ is the inner product on $\mathbb{R}^{n}$ which is given in the Cartesian coordinate system by $\langle x,y\rangle =\sum _{ i=1}^{n}x_{i}y_{i}$. In a standard coordinate system one has $\langle x,y\rangle =\sum _{i,j}g_{ij}x_{i}y_{j}$, where $g_{\mathit{ij}} =\langle e_{i},e_{j}\rangle$, and the metric tensor ((g _ij )) (cf. Chap. 7) is a positive-definite symmetric n × n matrix.

  In general, a Euclidean space is defined as a space, the properties of which are described by the axioms of Euclidean Geometry.

• Norm transform metric

  A norm transform metric is a metric d(x, y) on a vector space (V, | | . | | ), which is a function of | | x | | and | | y | . Usually, $V = \mathbb{R}^{n}$ and, moreover, $\mathbb{E}^{n} = (\mathbb{R}^{n},\vert \vert.\vert \vert _{2})$.

  Some examples are (p,q)-relative metric, M-relative metric and, from Chap. 19, the British Rail metric | | x | | + | | y | | for x ≠ y, (and equal to 0, otherwise), the radar screen metric min{1, | | x − y | | } and max{1, | | x − y | | } for x ≠ y. Cf. t-truncated and t-uniformly discrete metrics in Chap. 4.

• (p,q)-relative metric

  Let 0 < q ≤ 1, and $p \geq \max \{ 1 - q, \frac{2-q} {3} \}$. Let (V, | | . | | ) be a Ptolemaic space, i.e., the norm metric | | x − y | | is a Ptolemaic metric (cf. Chap. 1).

  The (p,q)-relative metric on (V, | | . | | ) is defined, for x or y ≠ 0, by

  $$\displaystyle{ \frac{\vert \vert x - y\vert \vert } {(\frac{1} {2}(\vert \vert x\vert \vert ^{p} + \vert \vert y\vert \vert ^{p}))^{\frac{q} {p} }}}$$

  (and equal to 0, otherwise). In the case of p = ∞, it has the form

  $$\displaystyle{ \frac{\vert \vert x - y\vert \vert } {(\max \{\vert \vert x\vert \vert,\vert \vert y\vert \vert \})^{q}}.}$$

  (p, 1)-, (∞, 1)- and the original (1, 1)-relative metric on $\mathbb{E}^{n}$ are called p-relative (or Klamkin–Meir metric ), relative metric and Schattschneider metric .

• M-relative metric

  Let f: [0, ∞) → (0, ∞) be a convex increasing function such that $\frac{f(x)} {x}$ is decreasing for x > 0. Let (V, | | . | | ) be a Ptolemaic space, i.e., | | x − y | | is a Ptolemaic metric.

  The M-relative metric on (V, | | . | | ) is defined by

  $$\displaystyle{ \frac{\vert \vert x - y\vert \vert } {f(\vert \vert x\vert \vert ) \cdot f(\vert \vert y\vert \vert )}.}$$

• Unitary metric

  The unitary (or complex Euclidean) metric is the l ₂ -metric on $\mathbb{C}^{n}$ defined by

  $$\displaystyle{\vert \vert x - y\vert \vert _{2} = \sqrt{\vert x_{1 } - y_{1 } \vert ^{2 } +\ldots +\vert x_{n } - y_{n } \vert ^{2}}.}$$

  For n = 1, it is the complex modulus metric $\vert x - y\vert = \sqrt{(x - y)\overline{(x - y)}}$ on the Wessel–Argand plane (cf. Chap. 12).

• L _p -metric

  An L _p -metric $d_{L_{p}}$, 1 ≤ p ≤ ∞, is a norm metric on $L_{p}(\Omega,\mathcal{A},\mu )$ defined by

  $$\displaystyle{\vert \vert f - g\vert \vert _{p}}$$

  for any $f,g \in L_{p}(\Omega,\mathcal{A},\mu )$. The metric space $(L_{p}(\Omega,\mathcal{A},\mu ),d_{L_{p}})$ is called the L _p -space (or Lebesgue space ).

  Here $\Omega$ is a set, and $\mathcal{A}$ is n σ-algebra of subsets of $\Omega$, i.e., a collection of subsets of $\Omega$ satisfying the following properties:

  • 1. $\Omega \in \mathcal{A}$;

  • 2. If $A \in \mathcal{A}$, then $\Omega \setminus A \in \mathcal{A}$;

  • 3. If $A = \cup _{i=1}^{\infty }A_{i}$ with $A_{i} \in \mathcal{A}$, then $A \in \mathcal{A}$.

  A function $\mu: \mathcal{A}\rightarrow \mathbb{R}_{\geq 0}$ is called a measure on $\mathcal{A}$ if it is additive, i.e., $\mu (\cup _{i\geq 1}A_{i}) =\sum _{i\geq 1}\mu (A_{i})$ for all pairwise disjoint sets $A_{i} \in \mathcal{A}$, and satisfies $\mu (\varnothing ) = 0$. A measure space is a triple $(\Omega,\mathcal{A},\mu )$.

  Given a function $f: \Omega \rightarrow \mathbb{R}(\mathbb{C})$, its L _p -norm is defined by

  $$\displaystyle{\vert \vert f\vert \vert _{p} = \left (\int _{\Omega }\vert f(\omega )\vert ^{p}\mu (d\omega )\right )^{\frac{1} {p} }.}$$

  Let $L_{p}(\Omega,\mathcal{A},\mu ) = L_{p}(\Omega )$ denote the set of all functions $f: \Omega \rightarrow \mathbb{R}$ ($\mathbb{C}$) such that | | f | |  _p  < ∞. Strictly speaking, $L_{p}(\Omega,\mathcal{A},\mu )$ consists of equivalence classes of functions, where two functions are equivalent if they are equal almost everywhere, i.e., the set on which they differ has measure zero. The set $L_{\infty }(\Omega,\mathcal{A},\mu )$ is the set of equivalence classes of measurable functions $f: \Omega \rightarrow \mathbb{R}$ ($\mathbb{C}$) whose absolute values are bounded almost everywhere.

  The most classical example of an L _p -metric is $d_{L_{p}}$ on the set $L_{p}(\Omega,\mathcal{A},\mu )$, where $\Omega$ is the open interval (0, 1), $\mathcal{A}$ is the Borel σ-algebra on (0, 1), and μ is the Lebesgue measure. This metric space is abbreviated by L _p (0, 1) and is called L _p (0,1)-space.

  In the same way, one can define the L _p -metric on the set C _[a, b] of all real (complex) continuous functions on [a, b]: $d_{L_{p}}(f,g) = (\int _{a}^{b}\vert f(x) - g(x)\vert ^{p}\mathit{dx})^{\frac{1} {p} }$. For p = ∞, $d_{L_{\infty }}(f,g) =\max _{a\leq x\leq b}\vert f(x) - g(x)\vert$. This metric space is abbreviated by C _[a, b] ^p and is called $C_{[a,b]}^{p}$ -space.

  If $\Omega = \mathbb{N}$, $\mathcal{A} = 2^{\Omega }$ is the collection of all subsets of $\Omega$, and μ is the cardinality measure ( i.e., μ(A) = | A | if A is a finite subset of $\Omega$, and $\mu (A) = \infty$, otherwise), then the metric space $(L_{p}(\Omega,2^{\Omega },\vert.\vert ),d_{L_{p}})$ coincides with the space l _p ^∞.

  If $\Omega = V _{n}$ is a set of cardinality n, $\mathcal{A} = 2^{V _{n}}$, and μ is the cardinality measure, then the metric space $(L_{p}(V _{n},2^{V _{n}},\vert.\vert ),d_{L_{ p}})$ coincides with the space l _p ⁿ.

• Dual metrics

  The l _p -metric and the l _q -metric, 1 < p, q < ∞, are called dual if $1/p + 1/q = 1$.

  In general, when dealing with a normed vector space (V, | | . | |  _V ), one is interested in the continuous linear functionals from V into the base field ($\mathbb{R}$ or $\mathbb{C}$). These functionals form a Banach space $(V ^{{\prime}},\vert \vert.\vert \vert _{V ^{{\prime}}})$, called the continuous dual of V. The norm $\vert \vert.\vert \vert _{V ^{{\prime}}}$ on $V ^{{\prime}}$ is defined by $\vert \vert T\vert \vert _{V ^{{\prime}}} =\sup _{\vert \vert x\vert \vert _{V }\leq 1}\vert T(x)\vert$.

  The continuous dual for the metric space l _p ⁿ ($l_{p}^{\infty }$) is $l_{q}^{n}$ ($l_{q}^{\infty }$, respectively). The continuous dual of l ₁ ⁿ ($l_{1}^{\infty }$) is $l_{\infty }^{n}$ ($l_{\infty }^{\infty }$, respectively). The continuous duals of the Banach spaces C (consisting of all convergent sequences, with l _∞ -metric) and C ₀ (consisting of the sequences converging to zero, with l _∞ -metric) are both naturally identified with $l_{1}^{\infty }$.

• Inner product space

  An inner product space (or pre-Hilbert space) is a metric space (V, | | x − y | | ) on a real (complex) vector space V with an inner product 〈x, y〉 such that the norm metric | | x − y | | is constructed using the inner product norm $\vert \vert x\vert \vert = \sqrt{\langle x, x\rangle }$.

  An inner product 〈, 〉 on a real (complex) vector space V is a symmetric bilinear (in the complex case, sesquilinear) form on V, i.e., a function $\langle,\rangle: V \times V \longrightarrow \mathbb{R}$ ($\mathbb{C}$) such that, for all x, y, z ∈ V and for all scalars α, β, we have the following properties:

  • 1. 〈x, x〉 ≥ 0, with 〈x, x〉 = 0 if and only if x = 0;

  • 2. $\langle x,y\rangle = \overline{\langle y,x\rangle }$, where the bar denotes complex conjugation;

  • 3. $\langle \alpha x +\beta y,z\rangle =\alpha \langle x,z\rangle +\beta \langle y,z\rangle$.

  For a complex vector space, an inner product is called also a Hermitian inner product, and the corresponding metric space is called a Hermitian inner product space.

  A norm | | . | | in a normed space (V, | | . | | ) is generated by an inner product if and only if, for all x, y ∈ V, we have: $\vert \vert x + y\vert \vert ^{2} + \vert \vert x - y\vert \vert ^{2} = 2(\vert \vert x\vert \vert ^{2} + \vert \vert y\vert \vert ^{2})$.

  In an inner product space, the triangle equality (Chap. 1) $\vert \vert x - y\vert \vert = \vert \vert x\vert \vert + \vert \vert y\vert \vert$, for x, y ≠ 0, holds if and only if $\frac{x} {\vert \vert x\vert \vert } = \frac{y} {\vert \vert y\vert \vert }$, i.e., x − y ∈ [x, y].

• Hilbert space

  A Hilbert space is an inner product space which, as a metric space, is complete. More precisely, a Hilbert space is a complete metric space (H, | | x − y | | ) on a real (complex) vector space H with an inner product 〈, 〉 such that the norm metric | | x − y | | is constructed using the inner product norm $\vert \vert x\vert \vert = \sqrt{\langle x, x\rangle }$. Any Hilbert space is a Banach space.

  An example of a Hilbert space is the set of all sequences $x =\{ x_{n}\}_{n}$ of real (complex) numbers such that $\sum _{i=1}^{\infty }\vert x_{i}\vert ^{2}$ converges, with the Hilbert metric defined by

  $$\displaystyle{(\sum _{i=1}^{\infty }(x_{ i} - y_{i})^{2})^{\frac{1} {2} }.}$$

  Other examples of Hilbert spaces are any L ₂ -space, and any finite-dimensional inner product space. In particular, any Euclidean space is a Hilbert space.

  A direct product of two Hilbert spaces is called a Liouville space (or line space, extended Hilbert space).

  Given an infinite cardinal number τ and a set A of the cardinality τ, let $\mathbb{R}_{a}$, a ∈ A, be the copies of $\mathbb{R}$. Let $H(A) =\{\{ x_{a}\} \in \prod _{a\in A}\mathbb{R}_{a}:\sum _{a}x_{a}^{2} <\infty \}$; then H(A) with the metric defined for $\{x_{a}\},\{y_{a}\} \in H(A)$ as

  $$\displaystyle{(\sum _{a\in A}(x_{a} - y_{a})^{2})^{\frac{1} {2} },}$$

  is called the generalized Hilbert space of weight τ.

• Erdös space

  The Erdös space (or rational Hilbert space) is the metric subspace of l ₂ consisting of all vectors in l ₂ with only rational coordinates. It has topological dimension 1 and is not complete. Erdös space is homeomorphic to its countable infinite power, and every nonempty open subset of it is homeomorphic to whole space.

  The complete Erdös space (or irrational Hilbert space) is the complete metric subspace of l ₂ consisting of all vectors in l ₂ the coordinates of which are all irrational.

• Riesz norm metric

  A Riesz space (or vector lattice) is a partially ordered vector space (V _Ri , ⪯) in which the following conditions hold:

  • 1. The vector space structure and the partial order structure are compatible, i.e., from x⪯ y it follows that $x + z\preceq y + z$, and from x ≻ 0, $a \in \mathbb{R}$, a > 0 it follows that ax ≻ 0;

  • 2. For any two elements x, y ∈ V _Ri , there exist the join $x \vee y \in V _{\mathit{Ri}}$ and meet x ∧ y ∈ V _Ri (cf. Chap. 10).

  The Riesz norm metric is a norm metric on V _Ri defined by

  $$\displaystyle{\vert \vert x - y\vert \vert _{\mathit{Ri}},}$$

  where | | . | |  _Ri is a Riesz norm on V _Ri , i.e., a norm such that, for any x, y ∈ V _Ri , the inequality | x | ⪯ | y | , where $\vert x\vert = (-x) \vee (x)$, implies | | x | |  _Ri  ≤ | | y | |  _Ri .

  The space (V _Ri , | | . | |  _Ri ) is called a normed Riesz space. In the case of completeness, it is called a Banach lattice.

• Banach–Mazur compactum

  The Banach–Mazur distance d _BM between two n-dimensional normed spaces (V, | | . | |  _V ) and (W, | | . | |  _W ) is defined by

  $$\displaystyle{\ln \inf _{T}\vert \vert T\vert \vert \cdot \vert \vert T^{-1}\vert \vert,}$$

  where the infimum is taken over all isomorphisms T: V → W. It is a metric on the set X ⁿ of all equivalence classes of n-dimensional normed spaces, where V ∼ W if and only if they are isometric. Then the pair (X ⁿ, d _BM ) is a compact metric space which is called the Banach–Mazur compactum.

• Quotient metric

  Given a normed space (V, | | . | |  _V ) with a norm | | . | |  _V and a closed subspace W of V, let $(V/W,\vert \vert.\vert \vert _{V/W})$ be the normed space of cosets $x + W =\{ x + w: w \in W\}$, x ∈ V, with the quotient norm $\vert \vert x + W\vert \vert _{V/W} =\inf _{w\in W}\vert \vert x + w\vert \vert _{V }$.

  The quotient metric is a norm metric on V∕W defined by

  $$\displaystyle{\vert \vert (x + W) - (y + W)\vert \vert _{V/W}.}$$

• Tensor norm metric

  Given normed spaces (V, | | . | |  _V ) and (W, | | . | |  _W ), a norm | | . | | _⊗ on the tensor product V ⊗ W is called tensor norm (or cross norm) if $\vert \vert x \otimes y\vert \vert _{\otimes } = \vert \vert x\vert \vert _{V }\vert \vert y\vert \vert _{W}$ for all decomposable tensors x ⊗ y.

  The tensor product metric is a norm metric on V ⊗ W defined by

  $$\displaystyle{\vert \vert z - t\vert \vert _{\otimes }.}$$

  For any z ∈ V ⊗ W, $z =\sum _{j}x_{j} \otimes y_{j}$, x _j  ∈ V, y _j  ∈ W, the projective norm (or π-norm) of z is defined by $\vert \vert z\vert \vert _{\mathit{pr}} =\inf \sum _{j}\vert \vert x_{j}\vert \vert _{V }\vert \vert y_{j}\vert \vert _{W}$, where the infimum is taken over all representations of z as a sum of decomposable vectors. It is the largest tensor norm on V ⊗ W.

• Valuation metric

  A valuation metric is a metric on a field $\mathbb{F}$ defined by

  $$\displaystyle{\vert \vert x - y\vert \vert,}$$

  where | | . | | is a valuation on $\mathbb{F}$, i.e., a function $\vert \vert.\vert \vert: \mathbb{F} \rightarrow \mathbb{R}$ such that, for all $x,y \in \mathbb{F}$, we have the following properties:

  • 1.  | | x | | ≥ 0, with | | x | | = 0 if and only if x = 0;

  • 2.  | | xy | | = | | x | | | | y | | ,

  • 3. $\vert \vert x + y\vert \vert \leq \vert \vert x\vert \vert + \vert \vert y\vert \vert$ (triangle inequality).

  If | | x + y | | ≤ max{ | | x | | , | | y | | }, the valuation | | . | | is called non-Archimedean. In this case, the valuation metric is an ultrametric. The simplest valuation is the trivial valuation $\vert \vert.\vert \vert _{\mathit{tr}}$: $\vert \vert 0\vert \vert _{\mathit{tr}} = 0$, and | | x | |  _tr  = 1 for $x \in \mathbb{F}\setminus \{0\}$. It is non-Archimedean.

  There are different definitions of valuation in Mathematics. Thus, the function $\nu: \mathbb{F} \rightarrow \mathbb{R} \cup \{\infty \}$ is called a valuation if ν(x) ≥ 0, ν(0) = ∞, $\nu (\mathit{xy}) =\nu (x) +\nu (y)$, and ν(x + y) ≥ min{ν(x), ν(y)} for all $x,y \in \mathbb{F}$. The valuation | | . | | can be obtained from the function ν by the formula | | x | | = α ^ν(x) for some fixed 0 < α < 1 (cf. p-adic metric in Chap. 12).

  The Kürschäk valuation | . |  _Krs is a function $\vert.\vert _{\mathit{Krs}}: \mathbb{F} \rightarrow \mathbb{R}$ such that | x |  _Krs  ≥ 0, | x |  _Krs  = 0 if and only if x = 0, $\vert \mathit{xy}\vert _{\mathit{Krs}} = \vert x\vert _{\mathit{Krs}}\vert y\vert _{\mathit{Krs}}$, and $\vert x + y\vert _{\mathit{Krs}} \leq C\max \{\vert x\vert _{\mathit{Krs}},\vert y\vert _{\mathit{Krs}}\}$ for all $x,y \in \mathbb{F}$ and for some positive constant C, called the constant of valuation. If C ≤ 2, one obtains the ordinary valuation | | . | | which is non-Archimedean if C ≤ 1. In general, any | . |  _Krs is equivalent to some | | . | | , i.e., | . |  _Krs ^p = | | . | | for some p > 0.

  Finally, given an ordered group (G, ⋅ , e, ≤ ) equipped with zero, the Krull valuation is a function $\vert.\vert: \mathbb{F} \rightarrow G$ such that | x | = 0 if and only if x = 0, | xy | = | x | | y | , and | x + y | ≤ max{ | x | , | y | } for any $x,y \in \mathbb{F}$. It is a generalization of the definition of non-Archimedean valuation | | . | | (cf. generalized metric in Chap. 3).

• Power series metric

  Let $\mathbb{F}$ be an arbitrary algebraic field, and let $\mathbb{F}\langle x^{-1}\rangle$ be the field of power series of the form $w =\alpha _{-m}x^{m} +\ldots +\alpha _{0} +\alpha _{1}x^{-1}+\ldots$, $\alpha _{i} \in \mathbb{F}$. Given l > 1, a non-Archimedean valuation | | . | | on $\mathbb{F}\langle x^{-1}\rangle$ is defined by

  $$\displaystyle{\vert \vert w\vert \vert = \left \{\begin{array}{ccc} l^{m},&\mbox{ if }& w\neq 0, \\ 0, &\mbox{ if }&w = 0.\end{array} \right.}$$

  The power series metric is the valuation metric | | w − v | | on $\mathbb{F}\langle x^{-1}\rangle$.