CarmineCortez said:Homework Statement
Do you just go through the conditions for norm spaces ie:
A normed vector space is a mathematical concept that combines the properties of a vector space (a set of objects that can be added and multiplied by scalars) with the notion of a norm (a function that assigns a length or magnitude to each vector). This allows for the measurement of distance, magnitude, and direction in a vector space.
L1, L2, and L-Infinity norms are different types of norms that can be used to measure the length or magnitude of a vector. L1 norm is also known as the "Manhattan distance" and measures the absolute value of each element in the vector. L2 norm is also known as the "Euclidean distance" and measures the square root of the sum of squared elements in the vector. L-Infinity norm measures the maximum absolute value of the elements in the vector.
To prove that L1, L2, and L-Infinity are norms, we need to show that they satisfy three properties: non-negativity, definiteness, and the triangle inequality. Non-negativity means that the norm of a vector is always greater than or equal to zero. Definiteness means that the norm of a vector is equal to zero only if the vector itself is zero. The triangle inequality states that the norm of the sum of two vectors is less than or equal to the sum of their individual norms. By demonstrating that L1, L2, and L-Infinity norms satisfy these properties, we can prove that they are indeed norms.
L1, L2, and L-Infinity norms are important in normed vector spaces because they allow us to measure the distance, magnitude, and direction of vectors in a systematic and consistent way. They are also used in many applications, such as optimization, data analysis, and machine learning. Furthermore, they have different properties and applications, making them useful in different scenarios.
Yes, norms can be extended to other types of vector spaces, such as complex vector spaces, function spaces, and sequence spaces. However, the specific definitions and properties of the norms may vary depending on the type of vector space. It is important to choose the appropriate norm for a given vector space in order to accurately measure its properties.