These informal notes deal with some very basic objects in functional analysis, including norms and seminorms on vector spaces, bounded linear operators, and dual spaces of bounded linear functionals in particular.

In this thesis, In this thesis, general structure of fuzzy normed linear spaces is examined and basic concepts of bounded and continuous operators defined on these spaces are studied. Firstly, fuzzy norm on finite-dimensional linear spaces is defined and by means of α -cut set some properties of this norm are mentioned. In the second part, generalizing the definition of fuzzy norm given on fini...

Based on ideas of R.W. Cross, a simplified proof is presented of the density invariance of certain operational quantities associated with bounded linear operators in normed vector spaces. Let X , Y denote normed linear spaces and let T : X → Y be a bounded linear operator. Let I(X) denote the collection of infinite dimensional subspaces of X . For any subspace M of X , let SM = {m ∈ M : ‖m‖ = 1...

It easy to see that every bounded linear operator is continuous. It is not hard to show that the converse is also true. The notions of bounded and continuous thereby coincide for linear operators acting between normed spaces. It is customary to prefer the terminology bounded linear operator over that of continuous linear operator. The reason for this preference is the fact that the hard part of...

Notation. ε is an arbitrarily small positive number. L(X,Y ) is the Banach space of bounded linear operators T : X → Y . B(X) is the closed unit ball {x ∈ X : ‖x‖ 6 1}. N := N ∪ {∞} is the one-point compactification of the discrete space N. All operators are bounded and linear. A compact (topological) space always means a compact Hausdorff space. (We sometimes add “Hausdorff” explicitly for emp...

What Is Spectral Theory? By spectral theory we mean the theory of structure of certain bounded linear operators on a Hilbert space. In a broader sense, the history of spectral theory goes way back to the nineteenth century, when the objects of study used to be infinite systems of linear equations and integral equations. The subject was revolutionized in the late 1920s by von Neumann, when he de...

The main aim of the present paper is to establish various sharp upper bounds for the Euclidean operator radius of an n-tuple of bounded linear operators on a Hilbert space. The tools used are provided by several generalizations of Bessel inequality due to Boas-Bellman, Bombieri, and the author. Natural applications for the norm and the numerical radius of bounded linear operators on Hilbert spa...

In this paper, concept of fuzzy continuous operator, bounded linear operator are introduced in strong [Formula: see text]-b-normed spaces and their relations studied. Idea norm is developed completeness BF(X,Y) established.

In this paper, we prove that the topological dual of the Banach space of bounded measurable functions with values in the space of nuclear operators, furnished with the natural topology, is isometrically isomorphic to the space of finitely additive linear operator-valued measures having bounded variation in a Banach space containing the space of bounded linear operators. This is then applied to ...

Abstract G-frames are generalized frames which include ordinary frames, bounded invertible linear operators, as well as many recent generalizations of frames, e.g., bounded quasi-projectors and frames of subspaces. G-frames are natural generalizations of frames and provide more choices on analyzing functions from frame expansion coefficients. We give characterizations of g-frames and prove that...