Norms and Inner Products

Part of the Undergraduate Texts in Mathematics book series (UTM)


In this chapter, we generalize to more abstract settings two key properties of \({\mathbb{R}^n}\): the Euclidean norm of a vector and the dot product of two vectors. The generalizations, norms and inner products, respectively, are set in a general vector space. Many of our applications will be set in this framework.

The basic notions of topology go through with almost no change in the definitions. However, some theorems can be quite different. For example, being closed and bounded is not sufficient to imply compactness in an infinite-dimensional normed vector space, such as C[a,b].


Hilbert Space Fourier Series Orthonormal Basis Normed Space Triangle Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag New York 2009

Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of WaterlooOntarioCanada
  2. 2.Department of MathematicsUniversity of Nebraska-LincolnLincolnUSA

Personalised recommendations