Mathematics Asked on January 5, 2022
Why is identity map on a separable Hilbert space not compact? False proof.
Let $e_n$ be the orthonormal basis. Then the projection map onto $H$ is defined by $sum(x,e_k)e_k$. What is stopping us from taking a finite part of this series. This gives us a compact operators, that converge to the identity map. Clearly something is wrong here not all Hilbert spaces are finite dimensional.
Beacause that sequence of operators does not converge to the identity: if $ninBbb N$,$$leftlVert e_{n+1}-sum_{j=1}^nlangle e_{n+1},e_krangle e_krightrVert=lVert e_{n+1}rVert=1$$and therefore$$leftlVertoperatorname{Id}-sum_{k=1}^nlanglecdot,e_krangle e_krightrVertgeqslant1.$$
Answered by José Carlos Santos on January 5, 2022
1 Asked on December 6, 2020 by jiexiong687691
conditional expectation expected value measure theory probability theory
1 Asked on December 6, 2020 by stranger
3 Asked on December 6, 2020 by trivial-math-is-difficult
combinations combinatorics discrete mathematics permutations probability
1 Asked on December 6, 2020 by riyasudheen-t-k
1 Asked on December 6, 2020 by mads-peter-balle
derivatives partial derivative stationary point systems of equations
2 Asked on December 6, 2020
4 Asked on December 6, 2020 by felipeuni
0 Asked on December 6, 2020 by twosigma
2 Asked on December 5, 2020 by student
1 Asked on December 5, 2020 by dr-suess-official
1 Asked on December 5, 2020 by curious-2-learn
3 Asked on December 5, 2020 by ramana
calculus definite integrals integration real analysis sequences and series
0 Asked on December 5, 2020 by moooose
4 Asked on December 5, 2020 by gray
1 Asked on December 5, 2020 by lucozade
2 Asked on December 5, 2020 by adam-b
2 Asked on December 5, 2020 by fakeraker-p
1 Asked on December 5, 2020
convergence divergence eigenvalues eigenvectors numerical methods solution verification
1 Asked on December 5, 2020 by user_9
Get help from others!
Recent Questions
Recent Answers
© 2023 AnswerBun.com. All rights reserved. Sites we Love: PCI Database, MenuIva, UKBizDB, Menu Kuliner, Sharing RPP, SolveDir