Chapter III. Topological Properties
... 12.B Generalization of 12.A. Let X be a connected space and f : X → R a continuous function. Then f (X) is an interval of R. 12.C Corollary. Let J ⊂ R be an interval of the real line, f : X → R a continuous function. Then f (J) is also an interval of R. (In other words, continuous functions map inte ...
... 12.B Generalization of 12.A. Let X be a connected space and f : X → R a continuous function. Then f (X) is an interval of R. 12.C Corollary. Let J ⊂ R be an interval of the real line, f : X → R a continuous function. Then f (J) is also an interval of R. (In other words, continuous functions map inte ...
Functional Analysis
... The map f 7→ kf kp is a seminorm, but not a norm; the subadditivity of k · kp is known as Minkowski’s inequality (see [17, Theorem 28.19] for its proof). If N := {f ∈ Lp (I) : kf kp = 0} then Lp (I) := Lp (I)/N is a Banach space, with norm [f ] 7→ [f ] p := kf kp . (A function lies in N if and on ...
... The map f 7→ kf kp is a seminorm, but not a norm; the subadditivity of k · kp is known as Minkowski’s inequality (see [17, Theorem 28.19] for its proof). If N := {f ∈ Lp (I) : kf kp = 0} then Lp (I) := Lp (I)/N is a Banach space, with norm [f ] 7→ [f ] p := kf kp . (A function lies in N if and on ...
Chapter 6 Manifolds, Tangent Spaces, Cotangent Spaces, Vector
... manifold and have good notions of curves, tangent vectors, differential forms, etc. The small drawback with the more general approach is that the definition of a tangent vector is more abstract. We can still define the notion of a curve on a manifold, but such a curve does not live in any given Rn, ...
... manifold and have good notions of curves, tangent vectors, differential forms, etc. The small drawback with the more general approach is that the definition of a tangent vector is more abstract. We can still define the notion of a curve on a manifold, but such a curve does not live in any given Rn, ...
weakly almost periodic flows - American Mathematical Society
... equicontinuous. A pair of points (x,y),x,y G X is proximal if for some net ta G T, x ■ta and y ■ta converge to the same point. A flow is distal if it has no nontrivial proximal pairs. It is well known that the flow (X, T) is almost periodic iff E(X) is a compact topological group and the elements of ...
... equicontinuous. A pair of points (x,y),x,y G X is proximal if for some net ta G T, x ■ta and y ■ta converge to the same point. A flow is distal if it has no nontrivial proximal pairs. It is well known that the flow (X, T) is almost periodic iff E(X) is a compact topological group and the elements of ...
More on Generalized Homeomorphisms in Topological Spaces
... (i) g is g.Λs -open, if f is continuous and surjective. (ii) f is g.Λs -open, if g is irresolute, pre-semi-closed and bijective. Proof. (i) Let A be an open set in Y . Since f −1 (A) is open in X, (g ◦ f )(f −1 (A)) is a g.Λs -set in Z and hence g(A) is g.Λs -set in Z. This implies that g is a g.Λs ...
... (i) g is g.Λs -open, if f is continuous and surjective. (ii) f is g.Λs -open, if g is irresolute, pre-semi-closed and bijective. Proof. (i) Let A be an open set in Y . Since f −1 (A) is open in X, (g ◦ f )(f −1 (A)) is a g.Λs -set in Z and hence g(A) is g.Λs -set in Z. This implies that g is a g.Λs ...
