List questions from banach-spaces

Published on 01 Apr 2026 - 14:57

115

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Brezis' exercise 6.20.2: prove that $\sigma (T) = \{0\}$

Published on 01 Apr 2026 - 14:57

63

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Brezis' exercise 6.20.3: give an explicit formula for $(T-\lambda I)^{-1}$

Published on 01 Apr 2026 - 14:56

35

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Brezis' exercise 6.20.4: determine $T^*$

Published on 28 Mar 2026 - 9:33

43

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Is the cardinality of $L^1[0,1]$ greater than $\frak c$?

Published on 25 Mar 2026 - 11:16

59

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Does there exist a normed space over $\mathbb R$ whose completion has strictly larger cardinality?

Published on 25 Mar 2026 - 12:46

43

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Brezis' exercise 6.21.1: prove that $N$ is finite-dimensional

Published on 28 Mar 2026 - 9:55

27

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Brezis' exercise 6.21.3: prove that there is $K_2 >0$ such that $K_2 d(u, N) \le p (u)$ for all $u \in V$

Published on 28 Mar 2026 - 11:35

15

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Brezis' exercise 6.22.3: construct an example in $E=\mathbb{R}^2$ such that $Q(E V(T)) \neq E V(Q(T))$ and $Q(\sigma(T)) \neq \sigma(Q(T))$

Published on 05 Mar 2026 - 9:36

75

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When weakly convergence implies the norm convergence?

Published on 23 Feb 2026 - 6:17

57

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Brezis' exercise 6.22.5: prove that $Q(E V(T)) = E V(Q(T))$ and $Q(\sigma(T)) = \sigma(Q(T))$

Published on 28 Mar 2026 - 9:33

98

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Let $T \in \mathcal L (E)$. Then $\|T^3\|^2 \le \|T^2\|^3 \, \|T\|^2$

Published on 31 Jul 2023 - 20:54

53

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If $B$ is a uniformly convex Banach space and $A\hookrightarrow B,$ is $A$ uniformly convex as well?

Published on 01 Aug 2023 - 8:14

79

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How to prove that $(\frac{a_n}{n})_n$ is convergent in $\mathbb R$?

Published on 01 Apr 2026 - 15:00

51

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Brezis' exercise 6.23.7: prove that $\|T^n\| \le \frac{1}{n!}$

Published on 01 Apr 2026 - 16:23

List Question

Brezis' exercise 6.20.2: prove that $0 \notin EV(T)$

Brezis' exercise 6.20.2: prove that $\sigma (T) = \{0\}$

Brezis' exercise 6.20.3: give an explicit formula for $(T-\lambda I)^{-1}$

Brezis' exercise 6.20.4: determine $T^*$

Is the cardinality of $L^1[0,1]$ greater than $\frak c$?

Does there exist a normed space over $\mathbb R$ whose completion has strictly larger cardinality?

Brezis' exercise 6.21.1: prove that $N$ is finite-dimensional

Brezis' exercise 6.21.3: prove that there is $K_2 >0$ such that $K_2 d(u, N) \le p (u)$ for all $u \in V$

Brezis' exercise 6.22.3: construct an example in $E=\mathbb{R}^2$ such that $Q(E V(T)) \neq E V(Q(T))$ and $Q(\sigma(T)) \neq \sigma(Q(T))$

When weakly convergence implies the norm convergence?

Brezis' exercise 6.22.5: prove that $Q(E V(T)) = E V(Q(T))$ and $Q(\sigma(T)) = \sigma(Q(T))$

Let $T \in \mathcal L (E)$. Then $\|T^3\|^2 \le \|T^2\|^3 \, \|T\|^2$

If $B$ is a uniformly convex Banach space and $A\hookrightarrow B,$ is $A$ uniformly convex as well?

How to prove that $(\frac{a_n}{n})_n$ is convergent in $\mathbb R$?

Brezis' exercise 6.23.7: prove that $\|T^n\| \le \frac{1}{n!}$

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