Metric Spaces Not Based on Inner Product

AI Thread Summary
A metric space can be defined without relying on an inner product, as demonstrated by the example of a set X containing two points A and B in the plane, with a distance function d that measures the distance between them. This example confirms that d satisfies the properties of a metric function. The discussion emphasizes the existence of alternative metric spaces that do not derive from vector inner products. Such spaces expand the understanding of metrics beyond traditional Euclidean contexts. The exploration of non-inner product metrics is crucial for broader applications in mathematics.
kthouz
Messages
188
Reaction score
0
Can somebody give me an other metric space that is not dependent on the inner product i mean which is not derived from the inner product between two vectors.
 
Physics news on Phys.org
Let (X,d) be a metric space, where X = {A, B}, with A and B two points in the plane, and d the distance between them. It is easily verified that d is a metrix function on X.
 

Thread 'Family of lines that are at a distance of 5 from the origin'

I picked up this problem from the Schaum's series book titled "College Mathematics" by Ayres/Schmidt. It is a solved problem in the book. But what surprised me was that the solution to this problem was given in one line without any explanation. I could, therefore, not understand how the given one-line solution was reached. The one-line solution in the book says: The equation is ##x \cos{\omega} +y \sin{\omega} - 5 = 0##, ##\omega## being the parameter. From my side, the only thing I could...
View full post »

Similar threads

Is the Proof for a Complex Inner Product Space Correct?
Replies
5
Views
1K
I Inner product vs dot/scalar product
Replies
4
Views
3K
I Shouldn't this definition of a metric include a square root?
Replies
8
Views
2K
I Inner and Outer Product of the Wavefunctions
Replies
8
Views
3K
I Gradient as vector vs differential one-form
Replies
38
Views
6K
I ##L^2## square integrable function Hilbert space
Replies
11
Views
3K
A Would it matter which inner product I choose in quantum mechanics?
Replies
4
Views
2K
I Inner products and wavefunctions
Replies
9
Views
1K
POTW A Modified Basis in an Inner Product Space
Replies
2
Views
2K
MHB Proving Non-Degeneracy of Euclidean Inner Products
Replies
8
Views
2K

Hot Threads

Recent Insights

Back
Top