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I would ask if anyone knows the métrqiue on the quotient space? ie, if one has a metric on a vector space, how can we calculate the metric on the quotient space of E?

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- Thread starter math6
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- #1

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I would ask if anyone knows the métrqiue on the quotient space? ie, if one has a metric on a vector space, how can we calculate the metric on the quotient space of E?

- #2

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First, what do you mean with quotient space?? Do you mean the quotient space as defined in topology?? In that case, the quotient of a metric space is not in general metrizable.

If you're working with a normed vector space V and a closed subset W, then V/W (as vector space quotient) does carry a norm which is given by

[tex]\|x+W\|=inf \{\|x+w\|~\vert~w\in W\}[/tex]

If you're working with a normed vector space V and a closed subset W, then V/W (as vector space quotient) does carry a norm which is given by

[tex]\|x+W\|=inf \{\|x+w\|~\vert~w\in W\}[/tex]

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and thank you very much.

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Here's some info: http://people.math.gatech.edu/~heil/6338/summer08/section6a.pdf

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For instance, ℝ

The formula that micromass gave computes the shortest distance from any point in the plane to any point in the "zero plane." This is the simplest, most canonical way to extend a norm on the original space to a norm on the quotient space.

- #6

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If you're working with a normed vector space V and a closed subset W, then V/W (as vector space quotient) does carry a norm which is given by

[tex]\|x+W\|=inf \{\|x+w\|~\vert~w\in W\}[/tex]

By "closed subset W" you mean linear subspace?

- #7

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And yes, I made a mistake in my post, I needed W to be both closed and a subspace. Just being closed is obviously not enough.

- #8

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And yes, I made a mistake in my post, I needed W to be both closed and a subspace. Just being closed is obviously not enough.

I see. I guess this is necessary only for infinite dimensional case, correct?

- #9

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I see. I guess this is necessary only for infinite dimensional case, correct?

Yes. Finite dimensional subspaces are automatically complete (and thus closed).

- #10

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Thank you for your answers.

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