Normed vector space

A normed vector space is a mathematical structure consisting of a vector space X and a norm ||·|| on X. The norm measures the size or length of vectors in X and satisfies certain properties, such as non-negativity, homogeneity, and the triangle inequality. A normed vector space can be viewed as a linear space, meaning it has the properties of addition and scalar multiplication, or as a metric space, meaning it has a distance function induced by the norm.From: Applied Functional Analysis [2018], The Communications Handbook [2018]

p Spaces

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Published in Kenneth Kuttler, Modern Analysis, 2017

Kenneth Kuttler

We see that ‖ ‖p would be a norm if ‖f‖p = 0 implied f = 0. If we agree to identify all functions in Lp that differ only on a set of measure zero, then ‖ ‖p is a norm and Lp is a normed vector space. We will do so from now on. Next we show that Lp is a complete normed vector space, a Banach space.

Signal Space

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Published in Jerry D. Gibson, Mobile Communications Handbook, 2017

Rodger E. Ziemer

which is called the natural metric for the space. A normed vector space may be viewed either as a linear space, a metric space, or both. Its elements may be interpreted as vectors or points, a case in point being ordinary three-dimensional geometry wherein we can identify points as vectors emanating from the origin to the point in question.

Modified reference tracking MPC for constrained linear systems

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Journal Information

Published in International Journal of Systems Science, 2018

Tito L. M. Santos

Notation. For a given real scalar λ and a given set X, then . The notation represents the interior of a set X and denotes the relative interior1 of a set X. A positive-definite matrix T is denoted as T>0. For a given matrix M, denotes its transpose. For a given symmetric matrix P>0, where P is a capital letter. A normed vector space is described by the pair where X is a vector space and is an arbitrary vector norm. The matrix denotes a null matrix with n rows and m columns. The vector denotes a null vector with n rows. The matrix I denotes the identity matrix with a suitable dimension. For a given differentiable function , the notation describes its gradient. represents the set of non-negative integers. Given two vectors , and a set , then .

Normed vector space

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Modified reference tracking MPC for constrained linear systems