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326.500-12, VL / UE Mathematics for AI III, Niels Lubbes et al., 2025W
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Let be defined as .
Determine whether is a subspace of under the usual vector addition and scalar multiplication.
Let be a function, and let be a -algebra over . Similar to one of the exercises, we prove is a -algebra over .
If there is (are) mistake(s) in the proof, select all corresponding answers. Select "The proof is correct.", if and only if 1) there are no errors in the proof, and 2) the proof is complete.
Proof:
(A) In order to prove is a -algebra, we prove all three defining properties.
(B) First, we show .
(B 1) For this, we note that , as is a -algebra over .
(B 2) Then, as is a function , it follows and therefore .
(C) Then, we show that for any we have , where we take the complement in .
(C 1) If , then - by definition of - there exists .
(C 2) As is a -algebra, from it follows , where the complement has to be taken in .
(C 3) By the definition of we observe that .
(C 4) Therefore, as we get .
(D) Finally, we have to show that for any sequence of sets the union .
(D 1) Take such a sequence , then there must be a corresponding sequence with .
(D 2) By being a -algebra, it follows that the union of the is contained in , i.e. there is .
(D 3) Therefore, also .
This concludes the proof.
For the following statements, decide whether they are true or false. Give a tick for each correct statement.
Let be the continuous function defined by . The Fourier series of is given by What can we say about the convergence of the Fourier series to on [0,1) ?
Click all true statements.