What are seminormed vector spaces?
Firstly, we must grasp an elemental, yet influential mathematical concept, namely the vector space. This concept has significant implications in a multitude of mathematical disciplines such as linear algebra, functional analysis, and quantum mechanics.
Note: You can learn more about vector spaces here.
Seminorms
Once we comprehend the concept of a vector space, we can enrich it by introducing the idea of a seminorm. Similar to a norm, a seminorm is a function that takes a vector and returns a real number. However, seminorms lack one property that a norm possesses - it's not always positive definite.
Normed Spaces | Seminormed Spaces | |
Definition | A vector space equipped with a norm. | A vector space equipped with a seminorm. |
Subadditivity / Triangle Inequality | A norm satisfies the triangle inequality: for any vectors x and y, | A seminorm also satisfies the triangle inequality: for any vectors x and y, |
Absolute Homogeneity | For any vector x and any scalar s in a normed space, the norm of the scalar multiplication of s and x equals the absolute value of s times the norm of x. This is mathematically represented as | In a seminormed space, the seminorm also satisfies absolute homogeneity. That is, for any vector x and any scalar s, the seminorm of the scalar multiplication of s and x equals the absolute value of s times the seminorm of x. This is mathematically represented as |
Nonnegativity | A norm is nonnegative: | A seminorm is also nonnegative: |
Positive Definiteness / Point-Separating | A norm is positive definite: | A seminorm need not be positive definite: |
Seminormed vector spaces
In a seminormed vector space, each vector is assigned a certain magnitude. However, distinct from a normed vector space, even vectors which are not zero can exhibit zero magnitude. This property of allowing vectors with zero magnitude makes seminormed vector spaces distinct and interesting.
Significance of seminormed vector spaces
Seminormed vector spaces play a crucial role as they bridge the gap between simple linear algebra and more complex analysis. They serve as a stepping stone toward the study of more intricate structures such as normed spaces, Banach spaces, and Hilbert spaces.
Conclusion
Seminormed vector spaces are fundamental to modern mathematics. They exemplify the structure and interconnection among various areas of mathematical study.
Review Quiz
Which of the following is a defining property of a seminorm?
Subadditivity / Triangle inequality
Positive definite
Point-separating
Absolute homogeneity
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