Abstract
The first chapter introduces the fundamental concepts and conclusions of functional analysis so that readers can have a foundation for going on reading this book successfully and can also understand notations used in the book. The arrangement of this chapter is as follows: The first section deals with normed linear spaces and inner product spaces which both provide a platform for further investigation; the second section introduces convex sets, and the third section considers convex functions; the last section of this chapter introduces semi-continuous functions. These are all necessary for research of set-valued mappings and differential inclusions which are two key concepts in this book.
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Notes
- 1.
At the notation \( C\left(\left[a,b\right],\kern0.5em \mathbb{R}\right) \), C means continuous, [a,b] is the domain and \(\mathbb{R} \) is the range. \( C\left(\left[a,b\right], \mathbb{R}^{n}\right) \) has a similar meaning.
- 2.
The integration is always in the meaning of Lebesgue integration if we do not give an illustration.
- 3.
The inner product can be defined as \( X\times X\to \mathbb{C} \). But in this book, we mainly consider the inner product in \( \mathbb{R} \).
- 4.
If \( {x}_1={x}_2 \), \( x={x}_1+\lambda \left({x}_2-{x}_1\right)\equiv {x}_1 \). The line degenerates to a point. In this book we do not consider the special case.
- 5.
The reader should divide the notation \( {A}_1\cup {A}_2 \) from \( {A}_1+{A}_2 \).
- 6.
It is possible that {x 1 n } and {x 2 n } are constant sequences.
- 7.
The requirement is equivalent to \( {x}_3\notin \left\{x;x=\left(1-\lambda \right){x}_1+\lambda {x}_2,\lambda \in \mathbb{R}\right\} \).
- 8.
By adding zeros, we can require that x and y are yielded by the same \( {x}_{n_i} \)’s.
- 9.
In general, the gradient is a row vector, but we consider \( \nabla f \) to be a column vector for unification.
- 10.
A matrix M > 0 means that the matrix is symmetric and positive definite. \( M\ge 0 \) means it is semi-positive definite.
- 11.
The reader should understand the meaning of the notation Df(x 0)(Ï…). It is emphasized that in Df(x 0)(Ï…) \( {x}_0\in {\mathbb{R}}^n \) is treated as a parameter, Ï… \( \in {\mathbb{R}}^n \) is the argument, \( {\partial}_{\upsilon } \) is the subdifferential related to the argument Ï….
- 12.
The notation \( \underset{x\to {x}_0}{ \lim\;\sup }f(x) \) should be understood as follows. Let δ be a small positive real number, and \( a\left(\delta \right)=\underset{x\in B\left({x}_0,\delta \right)}{ \sup }f(x) \). Then a(δ) is a monotonously decreasing function with the decreasing of δ. Hence, the limitation \( \underset{\delta \downarrow 0}{ \lim }a\left(\delta \right) \) exists. Then \( \underset{x\to {x}_0}{ \lim\;\sup }f(x)=\underset{\delta \downarrow 0}{ \lim }a\left(\delta \right)=\underset{\delta \downarrow 0}{ \inf}\underset{x\in B\left({x}_0,\delta \right)}{ \sup}\;f(x) \). The meaning of \( \underset{x\to {x}_0}{ \lim\;\inf }f(x) \) is similar.
- 13.
If f(x k ) is not convergent, then we can consider a subsequence \( \left\{{x}_{k_j}\right\} \) such that \( f\left({x}_{k_j}\right) \) is convergent. Hence for simplicity, we assume directly the sequence f(x k ) is convergent. The skill will be used frequently and we shall not give any explanation below.
- 14.
Recall that a function is normal if its effective domain is nonempty.
- 15.
The argument of \( {f}^{\ast}\left({x}^{\ast}\right) \) is \( {x}^{\ast } \), it is independent the x in f(x).
- 16.
Here we denote x for \( {x}^{\ast \ast } \).
References
Conway JB (1985) A course in functional analysis [M]. Springer, New York
de Bruim JCA, Doris A, van de Wouw et al (2009) Control of mechanical motion systems with non-collocation of actuation and friction: a Popov criterion approach for input-to-state stability and set-valued nonlinearities [J]. Automatica 45:405–415
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© 2016 Shanghai Jiao Tong University Press, Shanghai and Springer-Verlag Berlin Heidelberg
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Han, Z., Cai, X., Huang, J. (2016). Convex Sets and Convex Functions. In: Theory of Control Systems Described by Differential Inclusions. Springer Tracts in Mechanical Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49245-1_1
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