Appendix: Basic Terminology

Abstract

In this appendix, we present definitions of basic terminology used in the book for the reader’s convenience. For a given set W, $x\in W$ means that x is an element of W.

In this appendix, we present definitions of basic terminology used in the book for the reader’s convenience. For a given set W, $x\in W$ means that x is an element of W.

5.1 Convergence

• (1) Let M be a set. A real-valued function d defined on $M\times M$ is said to be a metric if

  • (i) $d(x,y)=0$ if and only if $x=y$ for $x,y\in M$;

  • (ii) (symmetry) $d(x,y)=d(y,x)$ for all $x,y\in M$;

  • (iii) (triangle inequality) $d(x,y)\leq d(x,z)+d(z,y)$ for all $x,y,z\in M$.

  Here, $X_1\times X_2$ denotes the Cartesian product of two sets $X_1$ and $X_2$ defined by

  $$\displaystyle \begin{aligned} X_1 \times X_2 := \left\{ (x_1,x_2) \bigm| x_i \in X_i\ \text{for}\ i=1,2 \right\}. \end{aligned}$$

  The set M equipped with a metric d is called a metric space and denoted by $(M,d)$ if one needs to clarify the metric. Let W be a product of metric spaces of $(M_i,d_i)$ ($i=1,\ldots ,m$), i.e.,

  $$\displaystyle \begin{aligned} W &= \prod_{i=1}^m M_i = M_1 \times\cdots\times M_m \\ &:= \left\{ (x_1,\ldots,x_m) \bigm| x_i \in M_i \ \text{for}\ i =1, \ldots, m \right\}. \end{aligned} $$

  This W is metrizable, for example, with a metric

  $$\displaystyle \begin{aligned} d(x,y) = \left( \sum_{i=1}^m d_i(x_i, y_i)^2\right)^{1/2} \end{aligned}$$

  for $x=(x_1,\ldots ,x_m)$, $y=(y_1,\ldots ,y_m)\in W$. If $M_i$ is independent of i, i.e., $M_i=M$, then we simply write W as $M^m$.

  A subset A of M is said to be open if for any $x\in A$ there is $\varepsilon >0$ such that the ball$B_\varepsilon (x)=\left \{y\in M\bigm | d(y,x)\leq \varepsilon \right \}$ is included in A. If the complement$A^c$ is open, then A is said to be closed. The complement $A^c$ is defined by

  $$\displaystyle \begin{aligned} A^c = M \backslash A := \left\{ x \in M \bigm| x \not\in A \right\}. \end{aligned}$$

  For a set A, the smallest closed set including A is called the closure of A and denoted by $\overline {A}$. Similarly, the largest open set included in A is called the interior of A and denoted by $\operatorname {int}A$ or simply by $\mathring {A}$. By definition, $A=\overline {A}$ if and only if A is closed, and $A=\mathring {A}$ if and only if A is open. The set $\overline {A}\backslash \mathring {A}$ is called the boundary of A and denoted by $\partial A$. For a subset B of a set A, we say that B is dense in A if $\overline {B}=A$. A set A in M is bounded if there is $x_0\in M$ and $R>0$ such that A is included in $B_R(x_0)$. For a mappingf from a set S to M (i.e., an M-valued function defined on S), f is said to be bounded if its image$f(S)$ is bounded in M, where

  $$\displaystyle \begin{aligned} f(S) = \left\{ f(x) \bigm| x \in S \right\}. \end{aligned}$$

• (2) Let V  be a real vector space (a vector space over the field $\mathbf {R}$). A nonnegative function $\|\cdot \|$ on V  is said to be a norm if

  • (i) $\|x\|=0$ if and only if $x=0$ for $x\in V$;

  • (ii) $\|cx\|=|c|\|x\|$ for all $x\in V$ and all $c\in \mathbf {R}$;

  • (iii) (triangle inequality) $\|x+y\| \leq \|x\|+\|y\|$ for all $x,y\in V$.

  The vector space V  equipped with a norm $\|\cdot \|$ is called a normed vector space and denoted by $\left (V,\|\cdot \|\right )$ if one needs to clarify the norm. By definition,

  $$\displaystyle \begin{aligned} d(x,y) = \| x-y \| \end{aligned}$$

  is a metric. A normed vector space is regarded as a metric space with the foregoing metric.

• (3) Let $\{z_j\}^\infty _{j=1}$ be a sequence in a metric space $(M,d)$. We say that $\{z_j\}^\infty _{j=1}$converges to $z\in M$ if for any $\varepsilon >0$ there exists a natural number $n=n(\varepsilon )$ such that $j\geq n(\varepsilon )$ implies $d(z,z_j)<\varepsilon$. In other words,

  $$\displaystyle \begin{aligned} \lim_{j \to 0} d(z,z_j) = 0. \end{aligned}$$

  We simply write $z_j\to z$ as $j\to \infty$, or $\lim _{j\to \infty }z_j=z$. If $\{z_j\}^\infty _{j=1}$ converges to some element, we say that $\{z_j\}^\infty _{j=1}$ is a convergent sequence.

• (4) Let f be a mapping from a metric space $(M_1,d_1)$ to another metric space $(M_2,d_2)$. We say that $f(y)$converges to $a\in M_2$ as y tends to x if for any $\varepsilon >0$ there exists $\delta =\delta (\varepsilon )>0$ such that

  $$\displaystyle \begin{aligned} d_2 \left(f(y), a\right) < \varepsilon \quad \text{if}\quad d_1(y,x) < \delta. \end{aligned}$$

  We simply write $f(y)\to a$ as $y\to x$ or $\lim _{y\to x}f(y)=a$. If

  $$\displaystyle \begin{aligned} \lim_{y \to x} f(y) = f(x), \end{aligned}$$

  then f is said to be continuous at $x\in M_1$. If f is continuous at all $x\in M_1$, then f is said to be continuous on $M_1$ (with values in $M_2$). The space of all continuous functions on $M_1$ with values in $M_2$ is denoted by $C(M_1, M_2)$.

• (5) Let $\{z_j\}^\infty _{j=1}$ be a sequence in a metric space $(M,d)$. We say that $\{z_j\}^\infty _{j=1}$ is a Cauchy sequence if for any $\varepsilon >0$ there exists a natural number $n=n(\varepsilon )$ such that $j,k\geq n(\varepsilon )$ implies $d(z_j,z_k)<\varepsilon$. It is easy to see that a convergent sequence is always a Cauchy sequence, but the converse may not hold. We say that the metric space $(M,d)$ is complete if any Cauchy sequence is a convergent sequence.

• (6) Let $\left (V,\|\cdot \|\right )$ be a normed vector space. We say that V  is a Banach space if it is complete as a metric space. The norm $\|\cdot \|$ is often written as $\|\cdot \|{ }_V$ to distinguish it from other norms if we use several norms. We simply write $z_j\to z$ in V  (as $j\to \infty$) if $\lim _{j\to \infty }\|z_j-z\|{ }_V=0$ and $z\in V$ for a sequence $\{z_j\}_{j=1}^\infty$. We often say that $z_j$ converges to zstrongly in V  (as $j\to \infty$) to distinguish this convergence from other weaker convergences discussed later.

• (7) Let V  be a real vector space. A real-valued function $\langle \cdot ,\cdot \rangle$ defined on $V\times V$ is said to be an inner product if

  • (i) $\langle x,x \rangle \geq 0$ for all $x\in V$;

  • (ii) $\langle x,x \rangle = 0$ if and only if $x=0$;

  • (iii) (symmetry) $\langle x,y \rangle =\langle y,x \rangle$ for all $x,y \in V$;

  • (iv) (linearity) $\langle c_1 x_1 + c_2 x_2, y \rangle =c_1\langle x_1, y \rangle + c_2\langle x_2, y \rangle$ for all $x_1, x_2, y \in V$, $c_1, c_2 \in \mathbf {R}$.

  By definition, it is easy to see that

  $$\displaystyle \begin{aligned} \|z\| = \langle z, z \rangle^{1/2} \end{aligned}$$

  is a norm. The space with an inner product is regarded as a normed vector space with the foregoing norm. If this space is complete as a metric space, we say that V  is a Hilbert space. The Euclidean space ${\mathbf {R}}^N$ is a finite-dimensional Hilbert space equipped with a standard inner product. It turns out that any finite-dimensional Hilbert space is “isomorphic” to ${\mathbf {R}}^N$. Of course, a Hilbert space is an example of a Banach space.

• (8) Let V  be a Banach space equipped with norm $\|\cdot \|$. Let $V^*$ denote the totality of all continuous linear function(al)s on V  with values in $\mathbf {R}$. (By the Hahn–Banach theorem, the vector space $V^*$ has at least one dimension. Incidentally, Mazur’s theorem in the proof of Lemma 1.19 in Sect. 1.2.3 is another application of the Hahn–Banach theorem.)

  The space $V^*$ is called the dual space of V . Let $\{z_j\}^\infty _{j=1}$ be a sequence in $V^*$. We say that $\{z_j\}^\infty _{j=1}$ converges to $z\in V^* \ *$-weakly if

  $$\displaystyle \begin{aligned} \lim_{j\to\infty} z_j(x) = z(x) \end{aligned}$$

  for any $x\in V$. We often write $z_j\overset {*}{\rightharpoonup } z$ in $V^*$ as $j\to \infty$. Such a sequence $\{z_j\}^\infty _{j=1}$ is called a $*$-weak convergent sequence. The dual space $V^*$ is equipped with the norm

  $$\displaystyle \begin{aligned} \|z\|{}_{V^*} := \sup \left\{ z(x) \bigm| \|x\| = 1,\ x \in V \right\} = \sup_{\|x\|=1}z(x). \end{aligned}$$

  The space $V^*$ is also a Banach space with this norm. Here, for a subset A in $\mathbf {R}$, by $a=\sup A$ we mean that a is the smallest real member that satisfies $a\geq x$ for any $a\in A$. In other words, it is the least upper bound of A. The notation $\sup$ is the abbreviation of the supremum. Similarly, $\inf A$ denotes the greatest lower bound of A, and it is the abbreviation of the infimum. If $\sup A=a$ with $a\in A$, we write $\max A$ instead of $\sup A$. The same convention applies to $\inf$ and $\min$.

  Since $V^*$ is a Banach space, there is a notion of convergence in the metric defined by the norm. To distinguish this convergence from $*$-weak convergence, we say that $\{z_j\}^\infty _{j=1}$ converges to zstrongly in $V^*$ if

  $$\displaystyle \begin{aligned} \lim_{j\to\infty} \|z_j-z\|{}_{V^*} = 0, \end{aligned}$$

  and it is simply written $z_j\to z$ in $V^*$ as $j\to \infty$. By definition, $z_j\to z$ implies $z_j \overset {*}{\rightharpoonup } z$, but the converse may not hold.

• (9) Let A be a subset of a metric space M. The set A is said to be (sequentially) relatively compact if any sequence $\{z_j\}^\infty _{j=1}$ in A has a convergent subsequence in M. If, moreover, A is closed, we simply say that A is compact. When A is compact, it is always bounded. When A is a subset of ${\mathbf {R}}^N$, it is well known as the Bolzano–Weierstrass theorem that A is compact if and only if A is bounded and closed. However, if A is a subset of a Banach space V , such an equivalence holds if and only if V  is of finite dimension. In other words, a bounded sequence of an infinite-dimensional Banach space may not have a (strongly) convergent subsequence.

  There is a compactness theorem (Banach–Alaoglu theorem) that says if $\{z_j\}^\infty _{j=1}$ in a dual Banach space $V^*$ is bounded, i.e.,

  $$\displaystyle \begin{aligned} \sup_{j\geq1}\|z_j\|{}_{V^*} < \infty, \end{aligned}$$

  then it has a $*$-weak convergent subsequence (Exercise 1.9).

• (10) Let V  be a Banach space and $V^*$ denote its dual space. Let $\{x_k\}^\infty _{k=1}$ be a sequence in V . We say that $\{x_k\}^\infty _{k=1}$ converges to $x\in V$weakly if

  $$\displaystyle \begin{aligned} \lim_{k\to\infty} z(x_k) = z(x) \end{aligned}$$

  for all $z\in V^*$. We often write $x_k \rightharpoonup x$ in V  as $k\to \infty$. Such a sequence is called a weak convergent sequence.

  If a Banach space W is a dual space of some Banach space V , say, $W=V^*$, there are two notions, weak convergence and $*$-weak convergence. Let $\{z_j\}_{j=1}^\infty$ be a sequence in W. By definition, $z_j \overset {*}{\rightharpoonup } z$ (in W as $j\to \infty$) means that $\lim _{j\to \infty }z_j(x)=z(x)$ for all $x\in V$ while $z_j \rightharpoonup z$ (in W as $j\to \infty$) means that $\lim _{j\to \infty }y(z_j)=y(z)$ for all $y\in W^*=(V^*)^*$.

  The space V  can be continuously embedded in $V^{**}=(V^*)^*$. However, V  may not be equal to $V^{**}$. Thus, weak convergence is stronger than $*$-weak convergence. If $V=V^{**}$, then both notions are the same. The space V  is called reflexive if $V=V^{**}$.

• (11) If V  is a Hilbert space, it is reflexive. More precisely, the mapping $x\in V$ to $z\in V^*$ defined by

  $$\displaystyle \begin{aligned} z(y) = \langle x,y \rangle, \quad y \in V \end{aligned}$$

  is a linear isomorphism from V  to $V^*$, which is also norm preserving, i.e., $\|z\|{ }_{V^*}=\|x\|$. This result is known as the Riesz–Fréchet theorem. Thus, the notions of weak convergence and $*$-weak convergence are the same.

• (12) Let f be a real-valued function in a metric space M. We say that f is lower semicontinuous at $x\in M$ if

  $$\displaystyle \begin{aligned} f(x) \leq \liminf_{y\to x} f(y) := \lim_{\delta\downarrow 0}\inf \left\{ f(y) \bigm| d(y,x)<\delta \right\}, \end{aligned}$$

  where $\lim _{\delta \downarrow 0}$ denotes the limit as $\delta \to 0$ but restricted to $\delta >0$. Even if f is allowed to take $+\infty$, the definition of the lower semicontinuity will still be valid. If f is lower semicontinuous for all $x\in M$, we simply say that f is lower semicontinuous on M. If $-f$ is lower semicontinuous, we say that f is upper semicontinuous.

• (13) Let $f=f(t)$ be a function of one variable in an interval I in $\mathbf {R}$ with values in a Banach space V . We say that f is right differentiable at $t_0\in I$ if there is $v\in V$ such that

  $$\displaystyle \begin{aligned} \lim_{h\downarrow 0} \left\| f(t_0 + h) - f(t_0) - vh \right\| \bigm/h = 0 \end{aligned}$$

  provided that $t_0+h\in I$ for sufficiently small $h>0$. Such v is uniquely determined if it exists and is denoted by

  $$\displaystyle \begin{aligned} v = \frac{\mathrm{d}^+ f}{\mathrm{d}t} (t_0). \end{aligned}$$

  This quantity is called the right differential of f at $t_0$. The function $t\mapsto \frac {\mathrm {d}^+f}{\mathrm {d}t}(t)$ is called the right derivative of f. The left differentiability is defined in a symmetric way by replacing $h\downarrow 0$ with $h\uparrow 0$. Even if both right and left differentials exist, they may be different. For example, consider $f(t)=|t|$ at $t_0=0$. The right differential at zero is 1, while the left differential at zero is $-1$. If the right and left differentials agree with each other at $t=t_0$, we say that f is differentiable at $t=t_0$, and its value is denoted by $\frac {\mathrm {d}f}{\mathrm {d}t}(t_0)$. The function $t\mapsto \frac {\mathrm {d}f}{\mathrm {d}t}(t)$ is called the derivative of f. If f depends on other variables, we write $\partial f/\partial t$ instead of $\mathrm {d}f/\mathrm {d}t$ and call the partial derivative of f with respect to t.

5.2 Measures and Integrals

• (1) For a set M, let $2^M$ denote the family of all subsets of H. We say that a function $\mu$ defined on $2^M$ with values in $[0,\infty ]$ is an (outer) measure if

  • (i) $\mu (\emptyset )=0$;

  • (ii) (countable subadditivity) $\mu (A)\leq \sum ^\infty _{j=1} \mu (A_j)$ if a countable family $\{A_j\}^\infty _{j=1}$coversA, where $A_j,A\in 2^M$. In other words, A is included in a union of $\{A_j\}^\infty _{j=1}$, i.e., a point of A must be an element of some $A_j$.

  Here, $\emptyset$ denotes the empty set.

• (2) A set $A\in 2^M$ is said to be $\mu$-measurable if

  $$\displaystyle \begin{aligned} \mu(S \cap A^c) + \mu(S \cap A) = \mu(S) \end{aligned}$$

  for any $S\in 2^M$. Let $M_0$ be a metric space. A mapping f from M to $M_0$ is said to be $\mu$-measurable if the preimage$f^{-1}(U)$ of an open set U of $M_0$ is $\mu$-measurable. Here,

  $$\displaystyle \begin{aligned} f^{-1}(U) := \left\{ x \in M \bigm| f(x) \in U \right\}. \end{aligned}$$

  A set A with $\mu (A)=0$ is called a $\mu$-measure zero set. If a statement $P(x)$ for $x\in M$ holds for $x\in M\backslash A$ with $\mu (A)=0$, we say that $P(x)$ holds for $\mu$-almost every$x\in M$ or shortly a.e. $x\in M$. In other words, P holds in M outside a $\mu$-measure zero set. In this case, we simply say that P holds almost everywhere in M.

  Let $\mathcal {M}$ be the set of all $\mu$-measurable sets. If we restrict $\mu$ just to $\mathcal {M}$, i.e., $\overline {\mu }=\left .\mu \right |{ }_{\mathcal {M}}$, then $\overline {\mu }$ becomes a measure on $\mathcal {M}$. Since in this book we consider $\mu (A)$ for a $\mu$-measurable set A, we often say simply a measure instead of an outer measure.

• (3) Let A be a subset of ${\mathbf {R}}^N$. Let $\mathcal {C}$ be a family of closed cubes in ${\mathbf {R}}^N$ whose faces are orthogonal to the $x_i$-axis for some $i=1,\ldots ,N$. In other words, $C\in \mathcal {C}$ means

  $$\displaystyle \begin{aligned} C = \left\{ (x_1,\ldots,x_N) \in {\mathbf{R}}^N \bigm| a_i \leq x_i \leq a_i+\ell\ (i=1,\ldots,n) \right\} \end{aligned}$$

  for some $a_i$, $\ell \in \mathbf {R}$. Let $|C|$ denote its volume, i.e., $|C|=\ell ^N$. We set

  $$\displaystyle \begin{aligned} \mathcal{L}^N(A) = \inf \left\{ \sum^\infty_{j=1}|C_j| \Biggm| \{C_j\}^\infty_{j=1}\ \text{covers}\ A \ \text{with}\ C_j \in \mathcal{C} \right\}. \end{aligned}$$

  It turns out that $\mathcal {L}^N(C)=|C|$; it is nontrivial to prove $\mathcal {L}^N(C)\geq |C|$. It is easy to see that $\mathcal {L}^N$ is an (outer) measure in ${\mathbf {R}}^N$. This measure is called the Lebesgue measure in ${\mathbf {R}}^N$. It can be regarded as a measure in the flat torus ${\mathbf {T}}^N=\prod ^N_{i=1}(\mathbf {R}/\omega _i\mathbf {Z})$. For a subset A of ${\mathbf {T}}^N$, we regard this set as a subset $A_0$ of the fundamental domain (i.e., the periodic cell$[0,\omega _1)\times \cdots \times [0,\omega _N)$). The Lebesgue measure of A is defined by $\mathcal {L}^N(A)=\mathcal {L}^N(A_0)$. Evidently, $\mathcal {L}^N({\mathbf {T}}^N)=\omega _1\cdots \omega _N$, which is denoted by $|{\mathbf {T}}^N|$ in the proof of Lemma 1.21 in Sect. 1.2.5.

• (4) In this book, we only use the Lebesgue measure. We simply say measurable when a mapping or a set is $\mathcal {L}^N$-measurable. Instead of writing $\mathcal {L}^N$-a.e., we simply write a.e. Let $\Omega$ be a measurable set in ${\mathbf {T}}^N$ or ${\mathbf {R}}^N$, for example, $\Omega ={\mathbf {T}}^N$. Let f be a measurable function on $\Omega$ with values in a Banach space V . Then one is able to define its integral over $\Omega$. When $V={\mathbf {R}}^m$, this integral is called the Lebesgue integral. In general, it is called the Bochner integral of f over $\Omega$. Its value is denoted by $\int _\Omega f\,\mathrm {d}\mathcal {L}^N$ or, simply, $\int _\Omega f\,\mathrm {d}x$; See, for example, [90, Chapter V, Section 5]. If $\Omega ={\mathbf {T}}^N$ and f is continuous, this agrees with the more conventional Riemann integral. For $p\in [1,\infty )$ and a general Banach space V , let $\tilde {L}^p(\Omega ,V)$ denote the space of all measurable functions f with values in V  such that

  $$\displaystyle \begin{aligned} \|f\|{}_p = \left( \int_\Omega \left\|f(x)\right\|{}^p \mathrm{d}x \right)^{1/p} \end{aligned}$$

  is finite. If $\|f\|{ }_p$ is finite, we say that f is pth integrable. If $p=1$, we simply say f is integrable. If $f\in \tilde {L}^1(\Omega ,V)$, we say that f is integrable in $\Omega$. We identify two functions $f,g\in \tilde {L}^p(\Omega ,V)$ if $f=g$ a.e. and define $L^p(\Omega ,V)$ from $\tilde {L}^p(\Omega ,V)$ by this identification. It is a fundamental result that $L^p(\Omega ,V)$ is a Banach space equipped with the norm $\|\cdot \|{ }_p$. When $V=\mathbf {R}$, we simply write $L^p(\Omega )$ instead of $L^p(\Omega ,V)$. The case $p=\infty$ should be handled separately. For a general Banach space V , let $\tilde {L}^\infty (\Omega ,V)$ denote the space of all measurable functions f with values in V  such that

  $$\displaystyle \begin{aligned} \|f\|{}_\infty = \inf \left\{ \alpha \Bigm| \mathcal{L}^N \left( \left\{ x \in \Omega \bigm| \left\| f(x) \right\|{}_V > \alpha \right\} \right) = 0 \right\} \end{aligned}$$

  is finite. By the same identification, the space $L^\infty (\Omega ,V)$ can be defined. This space $L^\infty (\Omega ,V)$ is again a Banach space. Key theorems in the theory of Lebesgue integrals used in this book include the Lebesgue dominated convergence theorem and Fubini’s theorem. Here, we give a version of the dominated convergence theorem.

Theorem 5.1

Let V  be a Banach space. Let $\{f_m\}_{m=1}^\infty$ be a sequence in $L^1(\Omega ,V)$ . Assume that there is a nonnegative function $\varphi \in L^1(\Omega )$ independent of m such that $\left \|f_m(x)\right \|{ }_V\leq \varphi (x)$ for a.e. $x\in \Omega$ . If $\lim _{m\to \infty }f_m(x)=f(x)$ for a.e. $x\in \Omega$ , then

$$\displaystyle \begin{aligned} \lim_{m\to\infty} \int_\Omega f_m(x)\, \mathrm{d}x = \int f(x)\,\mathrm{d}x. \end{aligned}$$

In other words,$\lim _{m\to \infty } \left \|\int _\Omega f_m\, \mathrm {d}x - \int _\Omega f(x)\,\mathrm {d}x\right \|{ }_V=0$.

• Usually, V  is taken as $\mathbf {R}$ or ${\mathbf {R}}^N$, but it is easy to extend to this setting. For basic properties of the Lebesgue measure and integrals, see for example a classical book of Folland [42]. We take this opportunity to clarify $*$-weak convergence in $L^p$ space. A basic fact is that $\left (L^p(\Omega )\right )^*=L^{p'}(\Omega )$ for $1\leq p<\infty$, where $1/p+1/p'=1$. Note that $p=\infty$ is excluded, but $(L^1)^*=L^\infty$. Since $L^p$ is reflexive for $1<p<\infty$, weak convergence and $*$-weak convergence agree with each other. Let us write a $*$-weak convergence in $L^\infty$ explicitly. A sequence $\{f_m\}$ in $L^\infty (\Omega )$$*$-weakly converges to $f\in L^\infty (\Omega )$ as $m\to \infty$ if and only if

  $$\displaystyle \begin{aligned} \lim_{m\to\infty} \int_\Omega f_m\varphi\, \mathrm{d}x = \int_\Omega f\varphi\, \mathrm{d}x \end{aligned}$$

  for all $\varphi \in L^1(\Omega$). For detailed properties of $L^p$ spaces, see, for example, [19, Chapter 4].

  For a Banach space V -valued $L^p$ function, we also consider its dual space. That is, we have

  $$\displaystyle \begin{aligned} \left(L^p(\Omega,V)\right)^* = L^{p'}(\Omega,V^*) \quad \text{for}\quad 1 \leq p < \infty \end{aligned}$$

  with $1/p+1/p'=1$. (This duality—at least for reflexive V —can be proved along the same line as in [19, Chapter 4], where V  is assumed to be $\mathbf {R}$. For a general Banach space V , see, for example, [35, Chapter IV].) We consider $*$-weak convergence in $L^\infty (\Omega ,V)$ with $V=L^q(U)$, $1<q\leq \infty$, where $\Omega$ is an open interval $(0,T)$ and U is an open set in ${\mathbf {T}}^N$ or ${\mathbf {R}}^N$ since this case is explicitly used in Chap. 2. A sequence $\{f_m\}$ in $L^\infty \left (\Omega ,L^q(U)\right )$$*$-weakly converges to $f\in L^\infty \left (\Omega ,L^q(U)\right )$ as $m\to \infty$ if and only if

  $$\displaystyle \begin{aligned} \lim_{m\to\infty}\int_0^T \!\!\!\! \int_U f_m(x,t)\varphi(x,t)\mathrm{d}x\, \mathrm{d}t = \int_0^T \!\!\!\! \int_U f(x,t)\varphi(x,t)\mathrm{d}x\, \mathrm{d}t \end{aligned}$$

  for $\varphi \in L^1\left (\Omega ,L^{q'}(U)\right )$. (Note that the space $L^p\left (\Omega , L^q(U)\right )$ is identified with the space of all measurable functions $\varphi$ on $\Omega \times U$ such that $\int _0^T \|\varphi \|{ }_{L^q(U)}^p(t)\, \mathrm {d}t<\infty$ or $\int _0^T\left ( \int _U \left |\varphi (x,t)\right |{ }^q \mathrm {d}x\right )^{p/q}\, \mathrm {d}t<\infty$ for $1\leq p,q<\infty$.)

• (5) Besides the basic properties of the Lebesgue integrals, we frequently use a few estimates involving $L^p$-norms. These properties are by now standard and found in many books, including [19]. For example, we frequently use the Hölder inequality

  $$\displaystyle \begin{aligned} \| fg \|{}_p \leq \|f\|{}_r \|g\|{}_q \end{aligned}$$

  with $1/p=1/r+1/q$ for $f\in L^r(\Omega ), g\in L^q(\Omega )$, where $p,q,r\in [1,\infty ]$. Here, we interpret $1/\infty =0$. In the case $p=1$, $r=q=2$, this inequality is called the Schwarz inequality. As an application, we have Young’s inequality for a convolution

  $$\displaystyle \begin{aligned} \|f*g\|{}_p \leq \|f\|{}_q \|g\|{}_r \end{aligned}$$

  for $f\in L^q({\mathbf {R}}^N)$, $g\in L^r({\mathbf {R}}^N)$ with $1/p=1/q+1/r-1$ and $p,q,r\in [1,\infty ]$; see, for example, [45, Chapter 4]. In this book, we use this inequality when ${\mathbf {R}}^N$ is replaced by ${\mathbf {T}}^N$.

• (6) In analysis, we often need an approximation of a function by smooth functions. We only recall an elementary fact. The space $C^\infty _c(\Omega )$ is dense in $L^p(\Omega )$ for $p\in [1,\infty )$; see, for example, [19, Corollary 4.23]. However, it is not dense in $L^\infty (\Omega )$.