Lipschitz continuity

In this note, we talk about the Lipschitz continuity of a function on a metric space. We followed wikipedia.

Def. Given two metric space X, Y. A function f: X \to Y is called Lipschitz continuous if there exists a real constant K \ge 0 such that for all x_1, x_2 \in X, d(f(x_1), f(x_2)) \le K d(x_1, x_2).

The smallest constant is sometimes called the (best) Lipschitz constant.

A function is called locally Lipschitz continuous if for every x \in X, there exists a neighborhood U of x such that f restricted ti U is Lipschitz continuous. Equivalently, if X is a locally compact metric space, then f is locally Lipschitz if and only if it is Lipschitz continuous on every compact subset of X. In spaces that are not locally compact, this is a necessary but not a sufficient condition.

More generally, a function f defined on X is said to be Holder continuous or to satisfy a Holder condition of order \alpha > 0 on X if there exists a constant M > 0 such that d(f(x), f(y)) \le M d(x, y)^{\alpha}.

Generalized gradient of Clarke


Def. The generalized gradient of f at x, denote \partial f(x) is the convex hull of the set of limits of the form \lim\nabla f(x + h_i) where h_i \to 0 (i \to \infty).
It follow that \partial f is a nonempty convex compact set.


Directional derivative

In multivariable calculus, there is  a notion of derivative, that is directional derivative.

Directional derivative is special derivative, follows a vector.

Definition. Let U \subset \mathbb{R}^m and U is open set. Let f: U \to \mathbb{R}^n. Suppose a \in U. Given v \in \mathbb{R}^m with v \ne 0, the directional of f at a corresponding to v is:

Df_v(a) = \lim\limits_{t \to 0}\dfrac{f(a + tu) - f(a)}{t}, if the limit exists.

Another definition: The directional derivative of f at a \in U corresponding to v is Df(a)\cdot v.

Theorem. Two above definitions are equivalent.


Germs of smooth functions

In maths, the notion of germ is used to definition “generic” of functions, in other hand, germ is used to say about mostly functions on topological spaces.

Consider to a point, we need to get information from that point, so we take a neighborhood of that point, assume that the point is x.

With a smooth function on manifold M, germ of smooth functions near x is a pair (U,f) such that f: U \to \mathbb{R} is smooth on local neighborhood U of x.

Two pairs (U,f) and (V,g) is equivalent if there is an open subset W \subset U \cap V such that f_{|_W} = g_{|_W}. So a germ of smooth function near x is an equivalent class of that pairs.

(to be cont.)

References: Wikipedia,

A classical result of Kadec and Pelczynski

A question of member of Mathoverflow:

In this question the norm of L^{P}[0,1] is denoted by \|.\|_p.

Let p and q be two arbitrary real numbers with 2<p<q.

> Assume that S is  a subvector space of L^{p}[0, 1] \bigcap L^{q}[0, 1] such that the identity operator \text{Id.}: (S, \|.\|_p) \to (S, \|.\|_q) is a bounded operator.  Does this implies that S is  a finite dimensional space?

If I am note mistaken, this is proved for p=2, q=\infty, by Grothendieck.

The answer of Bill Johnson on MO:

In fact, Kadec and Pelczynski proved that   a subspace of L_p, $2<p<\infty$, is closed in L_r for some r<p if and only if the subspace is isomorphic to a Hilbert space.

Kadec, M. I.; Pełczyński, A. Bases, lacunary sequences and complemented subspaces in the spaces Lp. Studia Math. 21 1961/1962 161–176.

Minimizing by Ekeland’s Variational Principle.

Minimizing by Ekeland’s Variational Principle.

Motivation of Ekeland’s paper “On the variational principle.” JMAA, 47. (1974). If f is a continous function on \mathbb{R}^n or Banach space X or complete Riemannian manifold \overline{M} and f has not minimum (of course f is bounded below) then we can choose a sequence (x_n) such that f(x_n) is “near” to the infimum of f.

We provide following Ekeland’s theorem with X is complete metric space

Theorem (Ekeland’s Variational Principle).  Let X be a complete metric space and F: X \to \mathbb{R} \cup \{+\infty\} a lower semi continous function, \ne +\infty, bounded from below. For every point u \in X satisfying \inf F \le F(u) \le \inf F + \epsilon and every \lambda > 0, there exists some point v \in X such that

F(v) \le F(u),

d(u,v) \le \lambda,

\forall w \ne v, F(w) > F(v) - (\epsilon/\lambda)d(v,w).

In part 5 of the paper, the author states an interesting result on complete Riemannian manifold

Proposition. Let F be a C^1 function on a complete Riemannian manifold \overline{M}. If F bounded from below, then, for every \epsilon \ge 0, there exists some point p_\epsilon \in \overline{M} such that

F(p_\epsilon) \le \inf F + \epsilon^2,

\|gradF(p_\epsilon)\|_{p_\epsilon} \le \epsilon.

Conclusion. p_\epsilon is the sequence of minimizer.