Metric Spaces Not Based on Inner Product

Click For Summary
SUMMARY

The discussion centers on identifying metric spaces that are not derived from inner products. A specific example provided is the metric space (X, d) where X consists of two points, A and B, in the plane, with d representing the distance between them. This example demonstrates that metric functions can exist independently of inner product definitions. The conversation emphasizes the fundamental properties of metric spaces, highlighting their versatility beyond inner product frameworks.

PREREQUISITES
  • Understanding of metric spaces and their properties
  • Familiarity with distance functions in mathematics
  • Basic knowledge of inner product spaces
  • Concept of point sets in topology
NEXT STEPS
  • Research different types of metric spaces, such as discrete and Euclidean spaces
  • Explore the properties of non-Euclidean metrics
  • Learn about the role of distance functions in topology
  • Investigate examples of metric spaces in real-world applications
USEFUL FOR

Mathematicians, students of topology, and anyone interested in advanced concepts of metric spaces and their applications in various fields.

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.
 

Similar threads

Is the Proof for a Complex Inner Product Space Correct?
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Inner product vs dot/scalar product
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Shouldn't this definition of a metric include a square root?
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad 2 interpretations of bra-ket expression: equal, & isomorphic, but...
  • · Replies 10 ·
Replies
10
Views
3K
Undergrad Inner and Outer Product of the Wavefunctions
  • · Replies 8 ·
Replies
8
Views
4K
Undergrad Gradient as vector vs differential one-form
  • · Replies 38 ·
2
Replies
38
Views
7K
Undergrad ##L^2## square integrable function Hilbert space
  • · Replies 11 ·
Replies
11
Views
3K
Graduate Would it matter which inner product I choose in quantum mechanics?
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Looking for learning resources for Clifford Algebra
  • · Replies 1 ·
Replies
1
Views
875
Undergrad Inner products and wavefunctions
  • · Replies 9 ·
Replies
9
Views
2K