Vector Extreme Value Theorem
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=Vector Extreme Value Theorem= | =Vector Extreme Value Theorem= | ||
| - | <m>F: \\bbR^n \\right \\bbR^ | + | <m>F: \\bbR^n \\right \\bbR^m</m> |
| - | <m>F</m> is continuous, <m>A \\subset \\bbR^ | + | <m>F</m> is continuous, <m>A \\subset \\bbR^n</m> compact subset |
| - | <m>\\right</m> | + | <m>\\right F(A)</m> is compact [i.e. F is bounded and attains its 'extreme' values] |
| - | + | ||
| - | And apparently, there's no analogue to the [[Intermediate Value Theorem]] in Vector Calculus. | + | And apparently, there's no analogue to the [[Intermediate Value Theorem]] in Vector Calculus. (CZ: As you have probably seen from the notes, that's as relative as the word ''intermediate''. The topological Extreme Value Theorem says that a continuous function maps a compact set to a compact set, and the Intermediate Value Theorem says that a continuous function maps a connected set to a connected set. Put them together and you get that a continuos function from <m>\\bbR</m> to <m>\\bbR</m> sends a closed interval to a closed interval.) |
And then I have these scribblings: | And then I have these scribblings: | ||
| - | <m> F(A) \\subset</m> | + | |
| + | If <m> F(A)\\subset \\bigcup{i \\in I}{}{U_i} </m> is an open covering of <m>F(A)</m> | ||
| + | |||
| + | then <m> A \\subset \\bigcup{i \\in I}{}{F^{-1}(U_i)} </m> | ||
| + | |||
| + | and since <m>F</m> is continuous, the inverse images are open sets, so we have an open covering of <m>A</m>. Since <m>A</m> is compact, there is a finite subcovering | ||
| + | |||
| + | <m> A \\subset \\bigcup{k=1}{n}{F^{-1}(U_i_k)} </m> | ||
| + | |||
| + | and therefore | ||
| + | |||
| + | <m>F(A) \\subset \\bigcup{k=1}{n}{(U_i_k)}</m> is a finite subcovering of the initial open covering. | ||
Current revision as of 00:17, 12 February 2006
[edit] Vector Extreme Value Theorem
<m>F: \\bbR^n \\right \\bbR^m</m> <m>F</m> is continuous, <m>A \\subset \\bbR^n</m> compact subset <m>\\right F(A)</m> is compact [i.e. F is bounded and attains its 'extreme' values]
And apparently, there's no analogue to the Intermediate Value Theorem in Vector Calculus. (CZ: As you have probably seen from the notes, that's as relative as the word intermediate. The topological Extreme Value Theorem says that a continuous function maps a compact set to a compact set, and the Intermediate Value Theorem says that a continuous function maps a connected set to a connected set. Put them together and you get that a continuos function from <m>\\bbR</m> to <m>\\bbR</m> sends a closed interval to a closed interval.)
And then I have these scribblings:
If <m> F(A)\\subset \\bigcup{i \\in I}{}{U_i} </m> is an open covering of <m>F(A)</m>
then <m> A \\subset \\bigcup{i \\in I}{}{F^{-1}(U_i)} </m>
and since <m>F</m> is continuous, the inverse images are open sets, so we have an open covering of <m>A</m>. Since <m>A</m> is compact, there is a finite subcovering
<m> A \\subset \\bigcup{k=1}{n}{F^{-1}(U_i_k)} </m>
and therefore
<m>F(A) \\subset \\bigcup{k=1}{n}{(U_i_k)}</m> is a finite subcovering of the initial open covering.
