Suppose f : A ? B is an isometry between metric spaces...

Name: Suppose f : A ? B is an isometry between metric spaces (A,dA) and (B,dB) with dB(f(a),f(b)) = dA(a,b) for all a,b ? A. If an isometry is surjective, prove that its inverse is continuous.
Uploaded: 2023-02-19T04:28:44-08:00
Duration: 4 min 56 s
Channel: Sri K
Description: Suppose f : A ? B is an isometry between metric spaces (A,dA) and (B,dB) with dB(f(a),f(b)) = dA(a,b) for all a,b ? A. If an isometry is surjective, prove that its inverse is continuous.

Suppose f : A ? B is an isometry between metric spaces (A,dA) and (B,dB) with dB(f(a),f(b)) = dA(a,b) for all a,b ? A. If an isometry is surjective, prove that its inverse is continuous.

Transcript

00:01 In this question, we have been given a function which is defined from set a to set b.

00:07 This is given as an isometry.

00:12 Okay, so this function is an isometry between the two metric spaces.

00:19 So a with the metrics represented by da, and we have another set, which is another metric space, so which is b comma db and db is the metric of b okay and the metric is defined as db of f a comma fb is equals to d a of a comma b okay and this is true for every ab belonging to the set a it says if f is surjective that means if the isometry is subjective then we have to prove that its inverse is continuous that is f inverse is continuous that we have to prove so let us see how we are going to do this correct i will consider two elements let's say a b in a then what will be my f of a so f of a will be f of b since it is an isometry and this will implies that d b of f of a f of b will be equal to zero which further implies d a of a comma b is equal to zero which implies a equals to b so this is from the definition of metric so this is by the definition of metric correct so now what we are getting f of a equals to f of p implies a equals to b so this will also imply that f is an injective function f is already given subjective okay so and it is already given that it is subjective so what does that mean f is bijective and we know that for a function to be invertible, f must be bisective.

02:28 So f is invertible.

02:32 From this we are getting.

02:34 So this will imply that inverse exists and it will be defined from b to a.

02:40 So i will say that this inverse function exists...

Question

Suppose f : A ? B is an isometry between metric spaces (A,dA) and (B,dB) with dB(f(a),f(b)) = dA(a,b) for all a,b ? A. If an isometry is surjective, prove that its inverse is continuous.

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