Correction to: Convex Analysis and Monotone Operator Theory in Hilbert Spaces

Correction to: H.H. Bauschke, P.L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics, https://doi.org/10.1007/978-3-319-48311-5

The original version of this book was inadvertently published without updating the following corrections in Chapters 1, 2, 3, 6–13, 17, 18, 20, 23, 24, 26, 29, 30 and back matter. These are corrected now.

For Figures 8.1, 13.1, 16.1, 18.1, 20.1, and 26.1—The usage of font/Symbol ${\mathcal {H}}$ invariably differs from universally adhered display standards throughout the text content and the same has been incorporated to resolve the issue.

Chapter 1

On page no. 9 (Fact 1.15, Eq. (1.28)) as per the recommendation additional commas in the mathematical function has been deleted and the parenthesis have been enlarged to accommodate the function well-within.

$\Big (\displaystyle\inf _{\substack {b\in A,\\ a\preccurlyeq b}}{\xi _b} \Big )_{a\in A}\quad\text{and}\quad\Big (\displaystyle\sup _{\substack {b\in A,\\ a\preccurlyeq b}}{\xi _b}\Big )_{a\in A}$

was replaced by:

$\left (\displaystyle\inf _{\substack {b\in A\\ a\preccurlyeq b}}{\xi _b} \right )_{a\in A}\quad\text{and}\quad\left (\displaystyle\sup _{\substack {b\in A\\ a\preccurlyeq b}}{\xi _b}\right )_{a\in A}$

On page no. 9, Lemma 1.16, Equation(1.29): The extra commas have been deleted.

$(\forall c\in A)\quad a\preccurlyeq c\quad \Rightarrow \quad \displaystyle\inf _{\substack {b\in A,\\ a\preccurlyeq b}}\xi _b + \inf _{\substack {b\in A,\\ a\preccurlyeq b}}\eta _b \leq \xi _c+\eta _c.$

was replaced by:

$(\forall c\in A)\quad a\preccurlyeq c\quad \Rightarrow \quad \displaystyle\inf _{\substack {b\in A\\ a\preccurlyeq b}}\xi _b + \inf _{\substack {b\in A\\ a\preccurlyeq b}}\eta _b \leq \xi _c+\eta _c.$

On page no. 9, Lemma 1.16, Equation(1.30): The extra commas have been deleted.

$\displaystyle\inf _{\substack {b\in A,\\ a\preccurlyeq b}}{\xi _b} + \inf _{\substack {b\in A,\\ a\preccurlyeq b}}{\eta _b} \leq \inf _{\substack {c\in A,\\ a\preccurlyeq c}}\,(\xi _c+\eta _c).$

was replaced by:

$\displaystyle\inf _{\substack {b\in A\\ a\preccurlyeq b}}{\xi _b} + \inf _{\substack {b\in A\\ a\preccurlyeq b}}{\eta _b} \leq \inf _{\substack {c\in A\\ a\preccurlyeq c}}\,(\xi _c+\eta _c).$

On page no. 19, (Second line after Eq. (1.55)) the inappropriate mathematical symbol “= ” is replaced with “≤”.

${\operatorname {diam}}\ B(x_n;\varepsilon _n)=2\varepsilon _n$

was replaced by:

${\operatorname {diam}}\, B(x_n;\varepsilon _n)\leq 2\varepsilon _n$

On page no. 21, Proof of Theorem 1.49 (4th line from the bottom of paragraph proof)—The missing period after the mathematical function has been put in place.

2δ ∕ 3 Then, using Example 1.48

was replaced by:

2δ ∕ 3. Then, using Example 1.48

Chapter 2

On page no. 31, Second line after Eq. (2.17)—The recommended change incorporated from “n × n” to “N × N”.

n × n

was replaced by:

N × N

On page no. 43, Fact 2.63—Insertion of mathematical function “T(U) ⊂ V ” and the correction of alphabetic representation of Gâteaux differentiable and Fréchet differentiable is successfully implemented.

let V  be a neighborhood of Tx, and let $R\colon V\to {\mathcal {K}}$. Suppose that R is Fréchet differentiable at x and that T is Gâteaux differentiable at Tx.

was replaced by:

let V  be a neighborhood of Tx such that T(U) ⊂ V , and let R: V → K. Suppose that T is Gâteaux differentiable at x and that R is Fréchet differentiable at Tx.

On page no. 44, Duplication of Exercise 2.6 (identical to Exercise 2.5) is eliminated and the rest of the exercises have been sequenced/re-numbered accordingly.

Chapter 3

On page no. 54, In the line preceding Example 3.18: Additional closing bracket has been introduced as suggested.

in Chapter 29.

was replaced by

in Chapter 29).

On page no. 66, Exercise 3.2: as per the instruction, the content “a nonempty set” is replaced with “a nonempty finite set”

a nonempty set

was replaced by

a nonempty finite set

Chapter 6

On page no. 113, Second line after Eq. (6.3): the Inappropriate mathematical symbol “= ” is replaced with “≤”.

∑_{i ∈ I} α _i = 1

was replaced by:

∑_{i ∈ I} α _i ≤ 1

Chapter 7

On page no. 135, Example 7.10: Inclusion of additional text is incorporated as recommended.

Let |||⋅||| be a norm on ${\mathcal {H}}$ and let

was replaced by:

Suppose that ${\mathcal {H}}$ is finite-dimensional, let |||⋅||| be a norm on ${\mathcal {H}}$, and let

Chapter 8

On page no. 142, Proof of Proposition 8.14(i): The Alphabetic representation “C” is replaced with “I” as per the instruction.

the convexity of ϕ on C

was replaced by:

the convexity of ϕ on I

On page no. 147, Example 8.31, Eq. (8.29): The incorrect variable “y” induced in the production is replaced with the appropriate representation “x”.

$y\in {{\mathbb R}_{++}}^N\;\text{and}\;y\in {{\mathbb R}_{++}}^N$

was replaced by:

$x\in {{\mathbb R}_{++}}^N\;\text{and}\;y\in {{\mathbb R}_{++}}^N$

On page no. 148, Example 8.33, last line of the proof: Appropriate Mathematical symbol is included as per the Instruction.

so is ϕ = f(⋅, 1).

was replaced by:

so is ϕ ^◇ = f(⋅, 1)

On page no. 155, Exercise 8.12(ii): Negative operator placed in front of the expression as suggested.

|x ^1∕p + 1|^p

was replaced by:

−|x ^1∕p + 1|^p

On page no. 155, Exercise 8.14, Eq. (8.53): The arguable content “and set” is replaced with “and assume that”.

and set

was replaced by:

and assume that

On page no. 156, Exercise 8.19: Recommended correction has been implemented.

0 ∈ C.

was replaced by:

0 ∈ri C.

Chapter 9

On page no. 169, Eq. (9.39): In the equation both occurrence pertaining to Lower-case omega “ω” were replaced with Upper-case Omega “Ω”.

μ(ω)

was replaced by:

μ( Ω)

On page no. 170, Eq. (9.40): The occurrence of Lower-case omega “ω” on the Right-hand side of the equation were replaced with Upper-case Omega “Ω”.

μ(ω)

was replaced by:

μ( Ω)

Chapter 10

On page no. 178, Example 10.6: Additional content displayed in question part of the example is removed as per the recommendation.

Then rec f is positively homogeneous and sublinear.

was replaced by:

Then rec f is sublinear.

Chapter 11

On page no. 191, Proposition 11.8(ii): As proposed the changes were implemented in both statement and proof part.

$C\cap {\operatorname {Argmin}} f$

was replaced by:

$({\mathrm{int}}\ C)\cap {\operatorname {Argmin}} f$

On page no. 196, Example 11.27: In the question part of the example the content “is separable” is replaced with suggested correction “is infinite-dimensional and separable.”

is separable.

was replaced by:

is infinite-dimensional and separable.

Chapter 12

On page no. 214, Eq. (12.37): Require confirmation from the author’s end as per his/her report the questionable data seems to be available in the Eq. (12.36) rather than the other Eq. (12.37).

$\displaystyle\inf _{\substack {x\in {\mathcal {H}},\\ Lx=y}}f(x)$

was replaced by:

$\displaystyle\inf _{\substack {x\in {\mathcal {H}}\\ Lx=y}}f(x)$

On page no. 214, Definition 12.34: Appropriate symbol representation is replaced as per the recommendation.

it is exact at a point $y\in {\mathcal {H}}$

was replaced by:

it is exact at a point $y\in {\mathcal {K}}$

On page no. 215, Example 12.35, Equation (12.39): The extra comma has been removed.

was replaced by

Chapter 13

On page no. 236, Exercise 13.3: Asterisk symbol in the expression is replaced as suggested.

The operation (α, f)↦α ∗ f

was replaced by:

The operation (α, f)↦α ⋆ f

On page no. 236, Exercise 13.4, Eq. (13.39): Addition symbol is introduced in the expression as indicated.

.

was replaced by:

.

Chapter 17

On page no. 302, Eq. (17.45): Recommended changes in the expression were Implemented.

N _C x ∩ B(0;ϕ′(d _C(x)))

was replaced by:

N _C x ∩ B(0;ϕ′(0))

Chapter 18

On page no. 324, Corollary 18.20, first sum in Eq. (18.40): Epsilon symbol is introduced in the expression as suggested.

$\displaystyle\sum _{i=1}^m\alpha _i{\operatorname {Prox}}_{f_i}$

was replaced by:

$\displaystyle\sum _{i\in I}\alpha _i{\operatorname {Prox}}_{f_i}$

On page no. 324, Fourth line after Eq. (18.40): Incorrect symbol representation in the expression is replaced.

$\nabla f=\sum _{i\in H}\alpha _i{\operatorname {Prox}}_{f_i}$

was replaced by:

$\nabla f=\sum _{i\in I}\alpha _i{\operatorname {Prox}}_{f_i}$

Chapter 20

On page no. 367, Exercise 20.1(iii): Recommended additional content is included per suggestion in sub-division (iii).

is differentiable and that

was replaced by:

is convex and differentiable, and that

Chapter 23

On page no. 404, Eq. (23.32): Second expression “⇔ p ∈ A(x − Up)” underneath the Eq. (23.32) has been deleted as per the instruction.

Chapter 24

On page no. 431, Eq. (24.81): Recommended changes implemented in the expression.

was replaced by:

$$\displaystyle \begin{aligned} \phi\colon{\mathbb R}&\to{\left]-\infty,+\infty\right]}\\ {}\xi&\mapsto \begin{cases} \dfrac{(1+\xi)\ln(1+\xi)+(1-\xi)\ln (1-\xi)-\xi^2}{2}&\text{if}\;\;|\xi|<1;\\ \ln(2)-1/2&\text{if}\;\;|\xi|= 1;\\ {}+\infty, &\text{if}\;\;|\xi|> 1.\\ \end{cases} \end{aligned} $$

On page no. 433, Example 24.51, Eq. (24.89): A period has been added at the end of equation

${\operatorname {Prox}}_{f}\colon {\mathcal {H}}\to {\mathcal {H}}\colon x\mapsto \begin {cases} x-\displaystyle{\frac {{\operatorname {Prox}}_{\phi ^*}\|x\|}{\|x\|}}x, &\text{if}\;\;\|x\|>\max \partial \phi (0);\\ 0,&\text{if}\;\;\|x\|\leq \max \partial \phi (0) \end {cases}$

was replaced by

${\operatorname {Prox}}_{f}\colon {\mathcal {H}}\to {\mathcal {H}}\colon x\mapsto \begin {cases} x-\displaystyle{\frac {{\operatorname {Prox}}_{\phi ^*}\|x\|}{\|x\|}}x, &\text{if}\;\;\|x\|>\max \partial \phi (0);\\ 0,&\text{if}\;\;\|x\|\leq \max \partial \phi (0). \end {cases}$

On page no. 433, In the line succeeding Eq. (24.89): The recommended changes incorporated

is a proximal thresholder

was replaced by

If ${\mathcal {H}}=\mathbb {R}$, then Prox_f is a proximal thresholder

On page no. 446, Exercise 24.7(ii): Sub-script for the function is retained as per recommendation.

${\operatorname {Prox}}{g}$

was replaced by:

${\operatorname {Prox}}_{g}$

Chapter 26

On page no. 470, Corollary 26.6: Negative operator has been introduced in both instances within the expression.

Then $x\in {\operatorname {zer}}(A+B)$, $u\in {\operatorname {zer}}(-A^{-1}\circ (-{{\operatorname {Id}}})+B^{-1})$, $(x,-u)\in {\operatorname {gra}} A$, and $(x,u)\in {\operatorname {gra}} B$.

was replaced by:

Then $x\in {\operatorname {zer}}(A+B)$, $-u\in {\operatorname {zer}}(-A^{-1}\circ (-{{\operatorname {Id}}})+B^{-1})$, $(x,u)\in {\operatorname {gra}} A$, and $(x,-u)\in {\operatorname {gra}} B$.

On page no. 489, Theorem 26.34, line after Eq. (26.97): The square option in the expression is removed as per suggestion.

let $\gamma \in \left ]0,1/\|L\|{ }^2\right [$

was replaced by:

let $\gamma \in \left ]0,1/\|L\|\right [$

Chapter 29

On page no. 536, Statement of Proposition 29.4: The proposed change to replace the symbol is implemented.

set

was replaced by:

set

On page no. 560, Statement of Exercise 29.23: The proposed change to include additional expression within the context is accommodated.

Show that $(\forall x\in {\operatorname {dom}} f\smallsetminus C)$

was replaced by:

Show that $(\forall x\in {\operatorname {dom}} f\smallsetminus C)dc(x)\leq ||x-x_0||(f(x)-\xi )/(f(x)-f(x_0))$

Chapter 30

On page no. 562, Eq. (30.8) and the sentence before: The proposed change to include additional expression within the context is accommodated.

and therefore that

$$\displaystyle \begin{aligned} (x_{n+1}-Tx_n)_{{n\in{\mathbb N}}}\;\text{is bounded.} \end{aligned}$$

was replaced by:

and therefore, in view of (i), that

$$\displaystyle \begin{aligned} ||x_{n+1}-Tx_n||=\lambda_n||x-Tx_n||\to 0. \end{aligned}$$

Back Matter

On page no. 592, Reference [82]: Additional bracket information in the reference context is removed.

Restricted Statistical Inference)

was replaced by:

Restricted Statistical Inference

On page no. 594, Reference [127]: Recommended change incorporated in the reference section.

examples (2016). http://arxiv.org/abs/1610.01552

was replaced by:

examples, Set-Valued Var. Anal., 26 (2018), pp. 247–264.

On page no. 595, Reference [131]: Recommended change incorporated in the reference section.

Appl. (2016). Published online 2016-12-15.

was replaced by:

Appl., 457 (2018), pp. 1283–1306.