Basic Framework

Abstract

In this chapter, we introduce the structure of a normed space. It involves the superposition of a metric structure on a linear space by means of a norm. Also, we introduce the structure of an inner product space and show that an inner product induces a special kind of norm. We consider the concept of orthogonality in the context of an inner product space. Our study of functional analysis will take place within these basic structures. In the last two sections, we investigate complete normed spaces (which are known as Banach spaces) as well as complete inner product spaces (which are known as Hilbert spaces). We consider many examples of Banach spaces and Hilbert spaces.

Exercises

• 2.1. Let p be a seminorm on a linear space X, and let $U:=\{x\in X:p(x)<1\}$. Then the set U is convex (that is, $(1-t)x+ty\in U$ whenever $x,y\in U$ and $t\in (0,1)$), absorbing (that is, for every $x \in X$, there is $r > 0$ such that $(x/r)\in U)$ and balanced (that is, $kx \in U$ whenever $x \in U$ and $k \in {\mathbb K}$ with $|k| \le 1$). (Compare Lemma 4.8.)

• 2.2. Let $m\ge 2$, and let $p_1,\ldots ,p_m$ be seminorms on a linear space X. Define

  $$ p(x):=\max \{p_1(x),\ldots ,p_m(x)\}\;\text { and } \; q(x):=\min \{p_1(x),\ldots ,p_m(x)\} $$

  for $x\in X$. If one of $p_1,\ldots ,p_m$ is a norm, then p is a norm on X. However, q may not be a seminorm even if each of $p_1,\ldots ,p_m$ is a norm on X.

• 2.3. The closure of $c_{00}$ in $(\ell ^1,\Vert \cdot \Vert _1)$ is $\ell ^1$, the closure of $c_{00}$ in $(\ell ^2,\Vert \cdot \Vert _2)$ is $\ell ^2$ and the closure of $c_{00}$ in $(\ell ^\infty ,\Vert \cdot \Vert _\infty )$ is $c_0$.

• 2.4. The inclusions $L^\infty ([0,1])\subset L^2([0,1])\subset L^1([0,1])$ are proper, but there is no inclusion relation among the normed spaces $L^1({\mathbb R}),\,L^2({\mathbb R}), L^\infty ({\mathbb R})$.

• 2.5. Let $X:=C([0,1])$, and let $\Vert \cdot \Vert _p$ be the induced norm on X as a subspace of $L^p([0,1]), \, p\in \{1,2,\infty \}$. Then the norm $\Vert \cdot \Vert _\infty$ is stronger than the norm $\Vert \cdot \Vert _2$, and the norm $\Vert \cdot \Vert _2$ is stronger than the norm $\Vert \cdot \Vert _1$, but any two of these norms are not equivalent.

• 2.6. Let $a<b$, and $X:= C^1([a,b])$. For $x\in X$, let $\Vert x\Vert ':=\max \{|x(a)|,\Vert x'\Vert _\infty \}$. The norm $\Vert \cdot \Vert '$ on X is equivalent to the norm $\Vert \cdot \Vert _{1,\infty }$ on X, which is stronger than but not equivalent to the norm $\Vert \cdot \Vert _\infty$ on X.

• 2.7. Let $(X,\Vert \cdot \Vert )$ be a normed space, Y be a linear space, and let F be a linear map from X onto Y. Define $q(y)\!:=\!\inf \{\Vert x\Vert :x\in X\text { and }F(x)=y\}$ for $y\in Y$. If $x\in X$ and $F(x)=y$, then $q(y)=\inf \{\Vert x+z\Vert :z\in Z(F)\}$. Consequently, q is a seminorm on Y. In fact, q is a norm on Y if and only if Z(F) is a closed subset of X.

• 2.8. Suppose E is a compact subset of a normed space X. If there are $x_0\in X$ and $r>0$ such that $U(x_0,r)\subset E$, then X is finite dimensional.

• 2.9. Let $E:=\{x\in \ell ^2:\sum _{j=1}^\infty |x(j)|^2\le 1\}$. Then E is closed and bounded, but not compact. The Hilbert cube $C:=\{x\in \ell ^2:j|x(j)|\le 1\text { for all }j\in {\mathbb N}\}$ is compact.

• 2.10. A normed space $(X, \Vert \cdot \Vert )$ is called strictly convex if $\Vert (x+y)/2\Vert <1$ whenever $x,y\in X,\,\Vert x\Vert =1=\Vert y\Vert$ and $x\ne y$. If a norm $\Vert \cdot \Vert$ on a linear space X is induced by an inner product, then $(X, \Vert \cdot \Vert )$ is strictly convex.

  If $p:=2$, then the normed spaces $(\ell ^p,\Vert \cdot \Vert _p)$ and $(L^p([0,1]),\Vert \cdot \Vert _p)$ are strictly convex, but if $p\in \{1,\infty \}$, then they are not strictly convex, and as a result, the norm $\Vert \cdot \Vert _p$ is not induced by any inner product.

• 2.11. Let X be an inner product space, and let $x,y\in X$. Then equality holds in the Schwarz inequality, that is, $|\langle {x},{y} \rangle |=\Vert x\Vert \Vert y\Vert$ if and only if $\langle {y},{y} \rangle x=\langle {x},{y} \rangle y$, and equality holds in the triangle inequality, that is, $\Vert x+y\Vert =\Vert x\Vert +\Vert y\Vert$ if and only if $\Vert y\Vert x=\Vert x\Vert y$.

• 2.12. (Parallelepiped law) Let X be an inner product space. Then

  $$ \Vert x+y+z\Vert ^2+ \Vert x+y-z\Vert ^2+ \Vert x-y+z\Vert ^2+ \Vert x-y-z\Vert ^2=4(\Vert x\Vert ^2+\Vert y\Vert ^2+\Vert z\Vert ^2) $$

  for all $x,y,z\in X$, that is, the sum of the squares of the lengths of the diagonals of a parallelepiped equals the sum of the squares of the lengths of its edges. If $z:=0$, then we obtain the parallelogram law (Remark 2.15).

• 2.13. Let X be a linear space over ${\mathbb K}$. Suppose a map $\langle {\cdot \,},{\cdot } \rangle :X\times X\rightarrow {\mathbb K}$ is linear in the first variable, conjugate-symmetric, and it satisfies $\langle {x},{x} \rangle \ge 0$ for all $x\in X$. Then $Z:=\{x\in X: \langle {x},{x} \rangle =0\}$ is a subspace of X. If we let $\langle \langle x+Z, y+Z\rangle \rangle :=\langle {x},{y} \rangle$ for $x+Z,\,y+Z$ in X/Z, then $\langle \langle \cdot \,, \cdot \rangle \rangle$ defines an inner product on X/Z. In particular, $|\langle {x},{y} \rangle |\le \langle {x},{x} \rangle ^{1/2}\langle {y},{y} \rangle ^{1/2}$ for all $x, y\in X$. (Compare Remark 2.2 and Exercise 4.37.)

• 2.14. (Polarization identity) Let $(X,\langle {\cdot \,},{\cdot } \rangle )$ be an inner product space. Then

  $$4\langle {x},{y} \rangle =\langle {x+y},{x+y} \rangle - \langle {x-y},{x-y} \rangle + i\langle {x+iy},{x+iy} \rangle - i\langle {x-iy},{x-iy} \rangle $$

  for all $x,y\in X$. (Note: The inner product $\langle {\cdot \,},{\cdot } \rangle$ is determined by the ‘diagonal’ subset $\{\langle {x},{x} \rangle :x\in X\}$ of $X\times X$. Compare Exercise 4.40.)

• 2.15. Let $\langle {\cdot \,},{\cdot } \rangle$ be an inner product on a linear space X. For nonzero $x,y\in X$, define the angle between x and y by

  $$\theta _{x,y} := \arccos \frac{\mathrm{Re\, }\langle {x},{y} \rangle }{\sqrt{\langle {x},{x} \rangle \langle {y},{y} \rangle }}, $$

  where $\arccos :[-1,1]\rightarrow [0,\pi ]$. Then $\theta _{x,y}$ satisfies the law of cosines

  $$ \Vert x\Vert ^2+\Vert y\Vert ^2-2\Vert x\Vert \,\Vert y\Vert \cos \theta _{x,y}=\Vert x-y\Vert ^2. $$

  In particular, $\theta _{x,y}=\pi /2$ if and only if $\Vert x\Vert ^2+\Vert y\Vert ^2=\Vert x-y\Vert ^2$.

• 2.16. Let X denote the linear space of all $m\times n$ matrices with entries in ${\mathbb K}$. For $M:=[k_{i,j}]$ and $N:=[\ell _{i,j}]$ in X, define $\langle {M},{N} \rangle :=\sum _{i=1}^m\sum _{j=1}^nk_{i,j}\overline{\ell _{i,j}}$. Then $\langle {\cdot \,},{\cdot } \rangle$ is an inner product on X. The induced norm

  $$ \Vert M\Vert _F:=\bigg (\sum _{i=1}^m\sum _{j=1}^n|k_{i,j}|^2\bigg )^{1/2} $$

  is known as the Frobenius norm of the matrix M. If $m=n$, and $I_n$ denotes the $n\times n$ identity matrix, then $\Vert I_n\Vert _F=\sqrt{n}$.

• 2.17. Let $w:=(w(1),w(2),\ldots )\in \ell ^\infty$ be such that $w(j)>0$ for all $j\in {\mathbb N}$, and define $\langle {x},{y} \rangle _w:=\sum _{j=1}^\infty w(j)x(j)\overline{y(j)}$ for $x,y\in \ell ^2$. Then $\langle {\cdot \,},{\cdot } \rangle _w$ is an inner product on $\ell ^2$. Also, the norm $\Vert \cdot \Vert _2$ on $\ell ^2$ is stronger than the norm $\Vert \cdot \Vert _w$ induced by the inner product $\langle {\cdot \,},{\cdot } \rangle _w$. Let $v:=(1/w(1),1/w(2),\ldots )$. The norms $\Vert \cdot \Vert _2$ and $\Vert \cdot \Vert _w$ are equivalent if and only if $v\in \ell ^\infty$, and then

  $$ \frac{1}{\Vert v\Vert _\infty }\Vert x\Vert _2^2\le \Vert x\Vert _w^2\le \Vert w\Vert _\infty \Vert x\Vert _2^2\quad \text {for all }x\in \ell ^2.$$

• 2.18. (Helmert basis) Let $m\ge 2$, and consider the usual inner product on ${\mathbb K}^m$. If $x_1:=e_1+\cdots +e_m$, and $x_n:=e_1-e_n$ for $n=2,\ldots ,m$, then the Gram–Schmidt procedure yields the basis $\{u_1,\ldots ,u_m\}$ for ${\mathbb K}^m$, where $u_1:=(e_1+\cdots +e_m)/\sqrt{m}$, and $u_n:=\big (e_1+\cdots +e_{n-1}-(n-1)e_n\big )/\sqrt{(n-1)n}$ for $n=2,\ldots ,m$. (Note: This basis is useful in Multivariate Statistics.)

• 2.19. (QR factorization) Let A be an $m\times n$ matrix such that the n columns of A form a linearly independent subset of ${\mathbb K}^m$. Then there is a unique $m\times n$ matrix Q, whose columns form an orthonormal subset of ${\mathbb K}^m$, and there is a unique $n\times n$ matrix R which is upper triangular and has positive diagonal entries, such that $A = QR$. The result also holds for an infinite matrix A whose columns form a linearly independent subset of $\ell ^2$.

• 2.20. Let $x\in C([-1,1])$, and suppose that x is not a polynomial. If $m\in {\mathbb N}$ and $p_0,p_1,\ldots ,p_m$ are the Legendre polynomials of degrees $0,1,\ldots , m$, then

  $$ \sum _{n=0}^m\bigg |\int _{-1}^1x(t)p_n(t)dt\bigg |^2 < \int _{-1}^1|x(t)|^2dt. $$

• 2.21. (Trigonometric polynomials on ${\mathbb R}$) For $r\in {\mathbb R}$, let $u_r(t):=e^{irt},\,t\in {\mathbb R}$. Let ${\mathbb K}:={\mathbb C}$, and let X be the subspace of $C({\mathbb R})$ spanned by $\{u_r:r\in {\mathbb R}\}$. For $p,q\in X$, define

  $$ \langle {p},{q} \rangle :=\lim _{T\rightarrow \infty }\frac{1}{2T}\int _{-T}^Tp(t)\overline{q(t)}dt. $$

  Then $\langle {\cdot \,},{\cdot } \rangle$ is an inner product on X, and $\{u_r:r\in {\mathbb R}\}$ is an uncountable orthonormal subset of X.

• 2.22. Let X and Y be linear spaces, and let $\langle {\cdot \,},{\cdot } \rangle _Y$ be an inner product on Y. Suppose $F:X\rightarrow Y$ is a linear map. For $x_1,x_2\in X$, define $\langle {x_1},{x_2} \rangle _X:=\langle {F(x_1)},{F(x_2)} \rangle _Y$. Then

  • (i) $\langle {\cdot \,},{\cdot } \rangle _X$ is an inner product on X if and only if the map F is one-one. (Compare Remark 2.2.)

  • (ii) Suppose the map F is one-one, and let $\{u_\alpha \}$ be an orthonormal subset of X. Then $\{F(u_\alpha )\}$ is an orthonormal subset of Y.

  • (iii) Suppose the map F is one-one and onto, and let $\{u_\alpha \}$ be a maximal orthonormal subset of X. Then $\{F(u_\alpha )\}$ is a maximal orthonormal subset of Y.

• 2.23. Let $\Vert \cdot \Vert$ and $\Vert \cdot \Vert '$ be equivalent norms on a linear space X. Then $(X,\Vert \cdot \Vert )$ is a Banach space if and only if $(X,\Vert \cdot \Vert ')$ is a Banach space.

• 2.24. $(c_0,\Vert \cdot \Vert _\infty )$ and $(c,\Vert \cdot \Vert _\infty )$ are Banach spaces. Also, if T is a metric space, then $(C_0(T),\Vert \cdot \Vert _\infty )$ is a Banach space.

• 2.25. Let $\Vert \cdot \Vert$ be a norm on the linear space X consisting of all polynomials defined on [a, b] with coefficients in ${\mathbb K}$. Then there is a sequence $(p_{n})$ in X such that $\sum ^{\infty }_{n=1} \Vert p_{n}\Vert < \infty ,$ but $\sum ^{\infty }_{n=1} p_{n}$ does not converge in X.

• 2.26. Let X be an inner product space.

  • (i) Every finite dimensional subspace of X is complete. (Note: This can be proved without using Lemma 2.8.)

  • (ii) The closed unit ball of X is compact if and only if X is finite dimensional. (Note: This can be proved without using Theorem 2.10.)

  • (iii) If X is complete, then it cannot have a denumerable (Hamel) basis. (Note: This can be proved without using Theorem 2.26.)

• 2.27. $\{ e_{1}, e_ {2}, \ldots \}$ is a Schauder basis for $c_0$. Also, $\{e_0, e_{1}, e_ {2}, \ldots \}$, is a Schauder basis for c, where $e_0:= (1,1,\ldots )$. If $u_n:=(e_{2n-1}+e_{2n})/2$ and $v_n:=(e_{2n-1}-e_{2n})/2$ for $n\in {\mathbb N}$, then $\{u_1,v_1,u_2,v_2,\ldots \}$ is a Schauder basis for $\ell ^1$. On the other hand, neither $\ell ^\infty$ nor $L^\infty ([a,b])$ has a Schauder basis.

• 2.28. For $j\in {\mathbb N}$, let $(X_{j},\langle {\cdot \,},{\cdot } \rangle _{j})$ be an inner product space over ${\mathbb K}$. Let $X:=\big \{(x(1), x(2), \ldots ) : x(j) \in X_{j}\text { for all }j\in {\mathbb N}\, \text { and } \sum ^{\infty }_{j=1} \langle {x(j)},{x(j)} \rangle _j <\infty \big \}.$ Define $\langle {x},{y} \rangle := \sum ^{\infty }_{j=1} \langle {x(j)},{y(j)} \rangle _{j}$ for $x, y \in X.$ Then X is a linear space over ${\mathbb K}$ with componentwise addition and scalar multiplication, and $\langle {\cdot \,},{\cdot } \rangle$ is an inner product on X. Further, X is a Hilbert space if and only if $X_{j}$ is a Hilbert space for each $j\in {\mathbb N}$. (Note: $X=\ell ^2$ if $X_j:={\mathbb K}$ for each $j\in {\mathbb N}$.)

• 2.29.

  • (i) Let $C^{k}([a,b])$ denote the linear space of all k times differentiable functions on [a, b] whose kth derivatives are continuous on [a, b]. For $x\in C^{k}([a,b])$, let $\Vert x\Vert _{k,\infty }:=\max \{\Vert x\Vert _\infty , \Vert x'\Vert _\infty ,\ldots , \Vert x^{(k)}\Vert _\infty \}$. Then $C^{k}([a,b])$ is a Banach space.

  • (ii) Let $W^{k,1}([a,b])$ denote the linear space of all $k-1$ times differentiable functions on [a, b] whose $(k-1)$th derivatives are absolutely continuous on [a, b]. For $x\in W^{k,1}([a,b])$, let $\Vert x\Vert _{k,1}:=\sum _{j=0}^k\Vert x^{(j)}\Vert _1$. Then the Sobolev space $W^{k,1}([a,b])$ of order (k, 1) is a Banach space.

  • (iii) Let $W^{k,2}([a,b])$ denote the linear space of all functions in $W^{k,1}([a,b])$ whose kth derivatives are in $L^2([a,b])$. For x and y in $W^{k,2}([a,b])$, let

    $$\langle {x},{y} \rangle _{k,2}:= \sum ^{k}_{j=0} \int _a^bx^{(j)}(t)\overline{y}^{(j)}(t)dm(t).$$

    Then the Sobolev space $W^{k,2}([a,b])$ of order (k, 2) is a Hilbert space.

• 2.30. Let $(x_n)$ be a sequence in a Hilbert space H such that the set $\{x_n:n\in {\mathbb N}\}$ is orthogonal. Then the following conditions are equivalent.

  • (i) The series $\sum _{n=1}^\infty x_n$ is summable in H.

  • (ii) There is $s\in H$ such that $\langle {s},{x_n} \rangle =\Vert x_n\Vert ^2$ for all $n\in {\mathbb N}$.

  • (iii) $\sum _{n=1}^\infty \Vert x_n\Vert ^2<\infty$.

• 2.31. For $n\in {\mathbb N}$, let $u_n:=(e_{3n-2}+e_{3n-1}+e_{3n})/\sqrt{3},\,v_n:=(e_{3n-2}- e_{3n-1})/\sqrt{2}$ and $w_n:=(e_{3n-2}+e_{3n-1}-2e_{3n})/\sqrt{6}$. Then $\{u_1,v_1,w_1,u_2,v_2,w_2,\ldots \}$ is an orthonormal basis for $\ell ^2$.

• 2.32. Let ${\mathbb K}:={\mathbb C}$ and $\omega :=e^{2\pi i/3}$. Let $u_n:=(e_{3n-2}+e_{3n-1}+e_{3n})/\sqrt{3},\,v_n:= (e_{3n-2}+\omega \,e_{3n-1}+\omega ^2e_{3n})/\sqrt{3}$ and $w_n:=(e_{3n-2}+\omega ^2e_{3n-1}+\omega \,e_{3n})/\sqrt{3}$ for $n\in {\mathbb N}$. Then $\{u_1,v_1,w_1,u_2,v_2,w_2,\ldots \}$ is an orthonormal basis for $\ell ^2$.

• 2.33. Let $H := L^{2}([-\pi ,\pi ])$. Let $u_0(t):=1/\sqrt{2\pi }$, and for $n\in {\mathbb N}$, let $u_{n}(t) := \cos nt/\sqrt{\pi }$, $v_{n}(t) := \sin nt/\sqrt{\pi },\, t\in [-\pi ,\pi ]$. Then $\{u_0, u_{1}, v_1, u_{2}, v_2, \ldots \}$ is an orthonormal basis for H. Let $x\in H$. If $2\pi a_{0}=\int _{-\pi }^\pi x(t)dm(t)$, and $\pi a_{n}= \int _{-\pi }^{\pi }x(t)\cos nt\,dm(t),\; \pi b_{n}= \int _{-\pi }^{\pi }x(t)\sin nt\,dm(t)$ for $n\in {\mathbb N}$, then $x(t) = a_{0} + \sum _{n=1}^{\infty }\big ( a_{n} \cos nt + b_{n} \sin nt \big ),$ where the series converges in the mean square, and

  $$\frac{1}{\pi } \int _{-\pi }^{\pi }|x(t)|^{2}dm(t) = 2|a_{0}|^{2} + \sum _{n=1}^{\infty } \left( |a_{n}|^{2} + |b_{n}|^{2} \right) . $$

• 2.34. Let $H := L^{2}([0,1]).$ Let $u_0(t):=1$, and for $n\in {\mathbb N}$, let $u_{n}(t) := \sqrt{2}\cos n\pi t$, $v_{n}(t) := \sqrt{2}\sin n\pi t,\, t\in [0,1]$. Then both $\{u_0, u_{1}, u_{2}, \ldots \}$ and $\{v_1,v_2,\ldots \}$ are orthonormal bases for H. Let $x\in H$. If $a_{0}:= \int _{0}^{1}x(t)dm(t)$, and $a_{n}:= 2 \int _{0}^{1}x(t)\cos n\pi t\, dm(t),\; b_{n}:=2\int _{0}^{1}x(t)\sin n\pi t \,dm(t)$ for $n\in {\mathbb N}$, then $x(t) = a_{0} + \sum _{n=1}^{\infty } a_{n} \cos n\pi t$, and $x(t) =\sum _{n=1}^{\infty } b_{n} \sin n\pi t$, where both series converge in the mean square.

• 2.35. Let $H := L^2([-1,1])$. Let $x_0(t):=1$, and for $n\in {\mathbb N}$, let $x_{n}(t) :=t^{3n}$ for t in $[-1,1]$. Suppose $v_0,v_1,v_2, \ldots$ are obtained by the orthonormalization of the linearly independent subset $\{x_0,x_1,x_2,\ldots \}$ of H. Then $\{v_0,v_1,v_2, \ldots \}$ is an orthonormal basis for H. Also, $v_{0}(t) := 1 / \sqrt{2},\, v_1(t):=\sqrt{7}\,t^3/\sqrt{2}$, and $v_{2}(t) := \sqrt{13}(7t^{6} - 1)/6\sqrt{2}$ for $t\in [-1,1]$.

• 2.36. Let $\{u_\alpha :\alpha \in A\}$ be an orthonormal basis for a Hilbert space H, and let $\{v_\beta :\beta \in B\}$ be an orthonormal basis for a Hilbert space G. Suppose $\phi$ is a one-one function from A onto B. For $x:=\sum _\alpha \langle {x},{u_\alpha } \rangle u_\alpha \in H$, define $F(x):=\sum _\alpha \langle {x},{u_\alpha } \rangle v_{\phi (\alpha )} \in G,$ where $\sum _\alpha$ denotes a countable sum. Then F is a Hilbert space isomorphism from H onto G.

• 2.37. Let E be a subset of a Hilbert space H. Then $E^{\perp \perp }$ is the closure of $\mathrm{span\,}E$.

• 2.38. Let G be a closed subspace of a Hilbert space H. For $x\in H$, let y be the orthogonal projection of x on G. Then y is the unique best approximation of x from G, that is, y is the unique element of G such that $\Vert x-y\Vert =d(x,G)$.

• 2.39.

  • (i) Let Y be a finite dimensional proper subspace of a normed space X. Then there is $x_1\in X$ with $\Vert x_1\Vert =1=d(x_1,Y)$.

  • (ii) Let G be a closed proper subspace of a Hilbert space H. Then there is $x_1\in H$ such that $\Vert x_1\Vert =1=d(x_1,G)$.

  (Compare Lemma 2.7 of Riesz.)

• 2.40. Let H be a Hilbert space with an inner product $\langle {\cdot \,},{\cdot } \rangle$, and let G be a closed subspace of H. For $x_1+G$ and $x_2+G$ in H/G, define $\langle \langle x_1+G, x_2+G\rangle \rangle :=\langle {x_1-y_1},{x_2-y_2} \rangle$, where $y_1$ and $y_2$ are the orthogonal projections of $x_1$ and $x_2$ on G respectively. Then $\langle \langle \cdot \,, \cdot \rangle \rangle$ is an inner product on H/G, and it induces the quotient norm on H/G. Further, H/G is a Hilbert space.