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Thread: Understanding Point Functions In Context of Metric Spaces

  1. #1
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    Understanding Point Functions In Context of Metric Spaces

    I am having trouble understanding the meaning of point functions. I know the mathematical definition but i don't think that i truly understand there true meaning.


    Point functions:


    Suppose $(X,d)$ is a metric space and $z∈X$. Then a nonnegative real function


    $x \rightarrow d(x,z)$ defined on $X$ is a point function at $z$.


    I have tried looking for examples w/ no luck.


    EDIT: Here is some more context to my question.


    http://s14.postimg.org/i1xzylwb5/0111152201.png
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  2. #2
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    Re: Understanding Point Functions In Context of Metric Spaces

    Hey iceblitzed.

    This basically looks like some sort of dirac delta type distance function from some given reference point - most likely by taking a single point and evaluating with respect to some metric.

    I guess you could think of each function with respect to a frame of reference. You basically have to translate the vector to some frame of reference before you apply the metric.

    The natural frame is with respect to the origin of the space (i.e. the zero vector) but it need not be that case. Every different point function treats a point as the origin and does the metric from that frame of reference.

    The idea of a frame of reference is the basis for linear and non-linear operators in differential geometry and linear algebra. Each basis has an operator to go from one basis to another and translations also fit this criteria.

    You could think of each point function as having a frame of reference and with enough of them you could build the metric by seeing how they all give some information (i.e. the point functions) to provide a way of getting an arbitrary metric - especially the ones where the points in question are close to the so called "point" functions.
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