Norms, lp norms, and the sup norm, Real Analysis II
In this lecture, we explore real normed linear spaces, similar to our previous discussion on inner product spaces. We start by generalizing the concept of measuring length in a vector space V using a norm. The norm must satisfy four properties: positivity, non-degeneracy, multiplicativity, and the triangle inequality.
(MA 426 Real Analysis II, Lecture 3)
We begin by defining these properties. Positivity ensures the norm is never negative. Non-degeneracy means the norm is zero if and only if the vector is the zero vector. Multiplicativity requires that scaling a vector scales its norm by the absolute value of the scalar. The triangle inequality states that the norm of the sum of two vectors is less than or equal to the sum of their norms.
Next, we explore different types of norms, starting with the L^p norms on Euclidean space. For any p greater than or equal to 1, the L^p norm is computed by raising each coordinate to the p-th power, summing them, and taking the p-th root of the sum. We examine the L^1 norm (taxi cab norm), the L^2 norm (Euclidean norm), and the sup norm.
We illustrate these norms by drawing unit circles in R^2 for each norm. The L^1 norm produces a diamond shape, the L^2 norm gives the familiar unit circle, and the sup norm results in a square. We observe how these shapes change as the value of p varies.
Finally, we demonstrate that in an inner product space, the inner product induces a norm. We verify the four properties of a metric—positivity, non-degeneracy, symmetry, and the triangle inequality—by leveraging the properties of the inner product from the previous lecture.
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