Recently, I am studying Mathematical Analysis. I found that there is a phrase called "Euclidean space" which common use $\mathbb{R}^n$ in MA. I know that a set V which satisfies 10 vector space axioms is called vector space. Then a vector space satisfies the inner product is called inner product space. A vector space which with Euclidean inner product called Euclidean space. Then the natural number $\mathbb{N}^n$ is not a vector space, but $\mathbb{R}^n,\,\mathbb{Q}^n,\,\mathbb{Z}^n,\,\mathbb{C}^n$ are.
I know that we can have measurement on inner product space, but the norm in Euclidean space also is a measurement on itself. Then metric space is also a measurement.
My question is what is the differences of the measurement between inner product space, Euclidean space, metric space? Need help for explanation, thanks.
The concepts of inner product space, Euclidean space, and metric space are fundamental in mathematics and have specific characteristics related to measurements and structures. Here's a breakdown of their differences in terms of measurements:
Inner Product Space:
Measurement: An inner product space is a vector space equipped with an inner product, which is a mathematical operation that quantifies the notion of "angle" and "length" for vectors. Properties: The inner product satisfies certain properties like linearity in the first argument, conjugate symmetry, and positive definiteness (which ensures non-negativity and vanishing only for the zero vector). Application: This structure allows for the definition of angles between vectors (via the inner product) and the derivation of norms (or lengths) of vectors based on the inner product.
Euclidean Space:
Measurement: A Euclidean space is a specific type of inner product space where the inner product (dot product) corresponds to the Euclidean inner product we are familiar with in classical geometry. Properties: Euclidean spaces have all the properties of an inner product space, but the inner product is specifically the dot product in n-dimensional real space $ℝ^n$. Application: In Euclidean spaces, the inner product (dot product) defines angles between vectors and the Euclidean norm (or length) of vectors. It's directly related to the geometrical concepts of distance, orthogonality, and angles in space.
Metric Space:
Measurement: A metric space is a set of points with a distance function (metric) defined between any two points. Properties: The distance function (metric) in a metric space satisfies specific properties like non-negativity, symmetry, and the triangle inequality. Application: While an inner product space or Euclidean space deals with notions of length, angles, and orthogonality, a metric space deals with distance. It provides a way to measure how far apart points are within the space, but it doesn't necessarily have notions of angles or lengths defined in terms of vectors. In summary, an inner product space focuses on angles and lengths defined by an inner product, while a Euclidean space is a specific type of inner product space where the inner product aligns with the traditional dot product in Euclidean geometry. On the other hand, a metric space deals solely with distances between points and doesn't inherently incorporate notions of angles or inner products. Each of these spaces provides different mathematical structures for understanding and measuring properties within mathematical objects.