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October 17th, 2010, 12:24 PM   #1
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Proving that a space is continuous, continuous at 0, and bdd

My problem is that I have a space

I need to prove the following are equivalent:
A. T is continuous.
B. T is continuous at 0.
C. T is bounded.

Thanks.
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October 18th, 2010, 02:14 PM   #2
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Re: Proving that a space is continuous, continuous at 0, and

Obvious.

To show we show that and then that .

Suppose is continuous at . Take and . Since is continuous at there exists such that for all such that . Hence, if we have that wich gives us that is continuous at . Thus is continuous.

Suppose is continuous. Consider the set . We have that is a compact set since the norm is a continous function from to , and is compact on . Hence since is continous \in A\}" /> is a compact set wich gives us that \in A\}" /> is a real bounded set. Take . We have that for all wich gives us that for all , . Hence for all since . Thus is bounded and then .


Suppose is bounded. Then there exists such that for all . Take and . Consider . If we have that . Hence is continuous at wich gives that is continuous.
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October 18th, 2010, 06:51 PM   #3
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Re: Proving that a space is continuous, continuous at 0, and

Quote:
Originally Posted by parasio

Just making a few corrections

Obvious.

To show we show that and then that .

Suppose is continuous at . Take and . Since is continuous at there exists such that for all such that . Hence, if we have that wich gives us that is continuous at . Thus is continuous.

Suppose is continuous. Consider the set . We have that is a compact set since the norm is a continous function from to , and is compact on . Hence since is continous \in A\}" /> is a compact set wich gives us that \in A\}" /> is a real bounded set. Take . We have that for all wich gives us that for all , . Hence for all since . Thus is bounded and then .


Suppose is bounded. Then there exists such that for all . Take and . Consider . If we have that . Hence is continuous at wich gives us that is continuous.
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October 18th, 2010, 07:33 PM   #4
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Re: Proving that a space is continuous, continuous at 0, and

Quote:
Originally Posted by parasio
Obvious.

To show we show that and then that .

Suppose is continuous at . Take and . Since is continuous at there exists such that for all such that . Hence, if we have that wich gives us that is continuous at . Thus is continuous.

Suppose is continuous. Consider the set . We have that is a compact set since the norm is a continous function from to , and is compact on . Hence since is continous \in A\}" /> is a compact set wich gives us that \in A\}" /> is a real bounded set. Take . We have that for all wich gives us that for all , . Hence for all since . Thus is bounded and then .


Suppose is bounded. Then there exists such that for all . Take and . Consider . If we have that . Hence is continuous at wich gives that is continuous.

How can we know that your proof :B=>A is correct??
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October 18th, 2010, 08:30 PM   #5
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Re: Proving that a space is continuous, continuous at 0, and

Suppose is continuous at . To show that is continuous you must show that is continuous at for all . Then you take an arbitrary and show that for all there exists such that for all such that . This is what I did here.

Take arbitrary. Now take arbitrary. Since is continuous at there exists such that for all such that . Hence, if we have that wich gives us that is continuous at . Thus is continuous.

There was a little mistake here, instead of "such that for all such that " it should be written such that " for all such that ". In the demonstration above it's already corrected.

If you have any problem just write me.
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October 19th, 2010, 03:57 PM   #6
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Re: Proving that a space is continuous, continuous at 0, and

Quote:
Originally Posted by parasio
Suppose is continuous at . To show that is continuous you must show that is continuous at for all . Then you take an arbitrary and show that for all there exists such that for all such that . This is what I did here.

Take arbitrary. Now take arbitrary. Since is continuous at there exists such that for all such that . Hence, if we have that wich gives us that is continuous at . Thus is continuous.

There was a little mistake here, instead of "such that for all such that " it should be written such that " for all such that ". In the demonstration above it's already corrected.

If you have any problem just write me.

To get |y-x|<? you have to use the substitution : y= y-x , infering that x=0 and thus proving continuity not for all ,x?E ,BUT FOR x=0 only.

Correct??
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October 19th, 2010, 07:19 PM   #7
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Re: Proving that a space is continuous, continuous at 0, and

You have that for all such that .

Now here's what I did. is a fixed point. For all we have that since is a linear space. Hence if you take such that you have from the above sentence that .

Any problem write again.
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October 20th, 2010, 04:12 PM   #8
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Re: Proving that a space is continuous, continuous at 0, and

Quote:
Originally Posted by paraiso
You have that for all such that .

Now here's what I did. is a fixed point. For all we have that since is a linear space. Hence if you take such that you have from the above sentence that .

Any problem write again.
That substitution works.

Now i have a question concerning the laws of logic involved in the above proof.

Since we know that in every mathematical proof we have the laws of logic applied on the theorems,definitions, or axioms involved in the proof and hence giving us the statements of the proof,which would you concur, are the laws of logic taking place in the above proof??
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October 20th, 2010, 05:36 PM   #9
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Re: Proving that a space is continuous, continuous at 0, and

Logic is present in everything you do in mathematics. I don't know the names of the laws applied here it's been a long time
since I studied logic for the last time.
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October 20th, 2010, 05:50 PM   #10
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Re: Proving that a space is continuous, continuous at 0, and

Quote:
Originally Posted by paraiso
Logic is present in everything you do in mathematics. I don't know the names of the laws applied here it's been a long time
since I studied logic for the last time.

You mean long time ago you could recognise the laws of logic involved in a mathematical analysis proof and now due to the lapse of time you cannot??
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