On Local Convexity in Hilbert Space
نویسندگان
چکیده
By means of his concept of local euclidean dimension of a set M at a point pÇ_M, Tietze reduces the proof of Theorem 1 to the case of locally convex continua with interior points and which coincide with the closure of their set of interior points. Tietze then proves the theorem for continua in Z22 and finally extends the proof to cover any Ek. Tietze's method does not seem to be applicable for sets in Hubert space. The following lines contain a simpler method of dealing with this problem which allows the establishment of Tietze's theorem in any real or complex normed vector space whether separable or not. For the sake of definiteness we state and prove our theorem for real Hubert space.
منابع مشابه
On Moduli of Convexity in Banach Spaces
Let X be a normed linear space, x ∈ X an element of norm one, and ε > 0 and δ(x,ε) the local modulus of convexity of X . We denote by ρ(x,ε) the greatest ρ≥ 0 such that for each closed linear subspace M of X the quotient mapping Q : X → X/M maps the open ε-neighbourhood of x in U onto a set containing the open ρ-neighbourhood of Q(x) in Q(U). It is known that ρ(x,ε) ≥ (2/3)δ(x,ε). We prove that...
متن کاملLocal characterization of strongly convex sets
Strongly convex sets in Hilbert spaces are characterized by local properties. One quantity which is used for this purpose is a generalization of the modulus of convexity δΩ of a set Ω. We also show that limε→0 δΩ(ε)/ε 2 exists whenever Ω is closed and convex.
متن کاملA new class of entanglement measures
Abstract We introduce new entanglement measures on the set of density operators on tensor product Hilbert spaces. These measures are based on the greatest cross norm on the tensor product of the sets of trace class operators on Hilbert space. We show that they satisfy the basic requirements on entanglement measures discussed in the literature, including convexity, invariance under local unitary...
متن کاملOn subdifferential in Hadamard spaces
In this paper, we deal with the subdierential concept onHadamard spaces. Flat Hadamard spaces are characterized, and nec-essary and sucient conditions are presented to prove that the subdif-ferential set in Hadamard spaces is nonempty. Proximal subdierentialin Hadamard spaces is addressed and some basic properties are high-lighted. Finally, a density theorem for subdierential set is established.
متن کاملA Noncommutative Convexity in C ∗ - Bimodules
Let A and B be C∗-algebras. We consider a noncommutative convexity in Hilbert A -B-bimodules, called A -B-convexity, as a generalization of C∗-convexity in C∗-algebras. We show that if X is a Hilbert A -B-bimodule, then Mn(X ) is a Hilbert Mn(A )-Mn(B)-bimodule and apply it to show that the closed unit ball of every Hilbert A -B-bimodule is A -B-convex. Some properties of this kind of convexity...
متن کاملConvexity and the Euclidean Metric of Space-Time
We address the reasons why the “Wick-rotated”, positive-definite, space-time metric obeys the Pythagorean theorem. An answer is proposed based on the convexity and smoothness properties of the functional spaces purporting to provide the kinematic framework of approaches to quantum gravity. We employ moduli of convexity and smoothness which are eventually extremized by Hilbert spaces. We point o...
متن کامل