Functional Analysis Notes (original) (raw)

1986

" 'The Journal of Functional Analysis' is dedicated to the broadening of the horizons of functional analysis. Accordingly, it encourages original research papers of high quality from all branches of science, provided the core and flavor are of a functional analytic character and the paper is in accordance with contemporary mathematical standards."-From the cover page of 'The Journal of Functional Analysis' (written in 1967.) 1 Hermann Amandus Schwarz (1843-1921), German mathematician, contemporary of Weierstrass, and known for his work in complex analysis. He is the one in many theorems in books on analytic functions. There are other two "Schwartz" (with a "t"): Laurent Schwartz (1915-2002), French mathematician, Fields Medal in 1950 for his work of distribution theory. Jack Schwartz (1930-2009), American mathematician, author of the famous book "Linear Operators". for all system of coefficients c 1 ,. .. , c n ∈ C, and all x 1 ,. .. , x n ∈ X. Given ϕ, set H 0 := ∑ finite c x δ x : x ∈ X, c x ∈ C = span C {δ x : x ∈ X} , and define a sesquilinear form on H 0 by ∑ c x δ x , ∑ d y δ y ϕ := ∑ c x d y ϕ (x, y). Note that ∑ c x δ x 2 ϕ := ∑ c x δ x , ∑ c x δ x ϕ = ∑ x,y c x c y ϕ (x, y) ≥ 0 by assumption. (All summations are finite.) However, •, • ϕ is in general not an inner product since the strict positivity axiom may not be satisfied. Hence one has to pass to a quotient space by letting N = f ∈ H 0 f , f ϕ = 0 , and set H := completion of the quotient space H 0 /N with respect to • ϕ. (The fact that N is really a subspace follows from (1.2).) H is a Hilbert space. Corollary 1.2. Let X be a set, and let ϕ : X × X → C be a function. Then ϕ is positive definite if and only if there is a Hilbert space H = H ϕ , and a function

Partial inner product spaces, a unifying concept in functional analysis - Theory and applications

2013

Many families of function spaces play a central role in analysis, in particular in signal processing (e.g. wavelet or Gabor analysis). Typical are Lp spaces, Besov spaces, amalgam spaces or modulation spaces. In all these cases, the parameter indexing the family measures the behavior (regularity, decay properties) of particular functions or operators. It turns out that all these space families are, or contain, scales or lattices of Banach spaces, which are special cases of partial inner product spaces (pip-spaces). In this context, it is often said that such families should be taken as a whole and operators, bases, frames on them should be defined globally, for the whole family, instead of individual spaces. In this paper, we shall give an overview of pip-spaces and operators on them, illustrating the results by space families of interest in mathematical physics and signal analysis. The interesting fact is that they allow a global definition of operators, and various operator classe...

Function Spaces – and How They Relate

2008

We’re not interested in a function here or a function there. We’re interested in the collection of all functions to R or to C. Why are we interested in these? Functions describe lots of things. For example, a function on a “physical body” could be used to describe the temperature at every point on the body. Functions to R or C are scalar-valued: they have their image in a field. We are often interested in vector-valued functions, for instance, vector fields on open sets in Euclidean space. However, the theory of vector-valued functions, in the finite-dimensional case, isn’t very different from that of scalar-valued functions, because, after all, the vector-valued function can be described by its scalar components. This raises a number of interesting points that we shall explore as we proceed. Another important point is that for most practical purposes, theorems which work for functions to R also work for functions to C. The main differences are: (1) For functions to R we use the abs...

A New Functional Analysis Approach

This paper’s aim is to propose an approach for finding the product’s functions-internal, functions between components and external functions regarding the product in relation with its environment and user - from a combined perspective, that of a functional analysis completed with useful and harmful function’s effects. Such a combined approach led to emphasis the harmful functions effects and further, trying to find solutions that avoid or eliminate these effects it can be obtained a better product conceptual design.

On some characterizations of inner product spaces

Journal of Mathematical Analysis and Applications, 2003

We study some characterizations of inner product spaces given in the literature. Among other things, we give an example showing that one of the characterizations given in the classical book of Amir (1986) is not correct.

New Functional Analysis Approach

ANNALS OF THE ORADEA UNIVERSITY. Fascicle of Management and Technological Engineering., 2011

This paper's aim is to propose an approach for finding the product's functions-internal, functions between components and external functions regarding the product in relation with its environment and userfrom a combined perspective, that of a functional analysis completed with useful and harmful function's effects. Such a combined approach led to emphasis the harmful functions effects and further, trying to find solutions that avoid or eliminate these effects it can be obtained a better product conceptual design.