Functional Analysis
نویسنده
چکیده
1. Metric and topological spaces A metric space is a set on which we can measure distances. More precisely, we proceed as follows: let X = ∅ be a set, and let d : X ×X → [0, ∞) be a map. Property 3 is called the triangle inequality. It says that a detour via z will not give a shortcut when going from x to y. The notion of a metric space is very flexible and general, and there are many different examples. We now compile a preliminary list of metric spaces. Example 1.1. If X = ∅ is an arbitrary set, then d(x, y) = 0 x = y 1 x = y defines a metric on X. This example does not look particularly interesting, but it does satisfy the requirements from Definition 1.1. Example 1.2. X = C with the metric d(x, y) = |x−y| is a metric space. X can also be an arbitrary non-empty subset of C, for example X = R. In fact, this works in complete generality: If (X, d) is a metric space and Y ⊂ X, then Y with the same metric is a metric space also.
منابع مشابه
Estimation of Climate Zone Effects on Iranian Temperature, Humidity, and Precipitation using Functional Analysis of Covariance
Functional Data Analysis (FDA) has recently made considerable progress because of easier access to the data that are essentially in the form of curves. Although functional modeling of Iranian precipitation based on temperature or humidity was done before, here we use functional analysis of variance and covariance to analyze the weather data collected randomly from Iranian weather stations in 20...
متن کاملPersian Handwriting Analysis Using Functional Principal Components
Principal components analysis is a well-known statistical method in dealing with large dependent data sets. It is also used in functional data for both purposes of data reduction as well as variation representation. On the other hand "handwriting" is one of the objects, studied in various statistical fields like pattern recognition and shape analysis. Considering time as the argument,...
متن کاملProviding comprehensive control chart for monitoring of linear and nonlinear profiles using functional data analysis.
Considering profiles as functional variables, two control charts are proposed for their monitoring in phase II. Due to its conformity with the nature of real-world profiles, applying functional model leads to proposed control charts obtained through functional data analysis techniques with desired features. These include simplicity in calculation and possibility of using them for different prof...
متن کاملGeneralization of Canonical Correlation Analysis from Multivariate to Functional Cases and its related problems
In multivariate cases, the aim of canonical correlation analysis (CCA) for two sets of variables x and y is to obtain linear combinations of them so that they have the largest possible correlation. However, when x and y are continouse functions of another variable (generally time) in nature, these two functions belong to function spaces which are of infinite dimension, and CCA for them should b...
متن کاملFunctional Evaluation of Mazandaran Province Cities Using Factor Analysis, Numerical Taxonomy and Cluster Analysis
متن کامل
Effect of home-based exercise on functional ability of hemodialysis patients: a systematic review and meta-analysis
Background: hemodialysis patients suffer from impaired functional ability. Several pilot studies have been conducted concerning the effect of home-based exercise on functional ability of hemodialysis patients; however, there have been observed some contradictions between the results of these studies. The aim of this study was, therefore, to determine the effects of home-based exercise on functi...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2009