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### Is $\text{Id}-T$ always invertible when $\lim_{n\to\infty} T^n = 0$?

Injective: Let us first prove that $I-T$ is injective. For this we check that the kernel is trivial. Let $v$ be in the kernel of $I-T$, i.e. $(I-T)v=0$, then $v=Iv=Tv$. However, this cannot be as ...
• 20.3k
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### Prove that the sequence $(A_n(f))$ is convergent in the normed space $C[0,1]$. Prove that the sequence $(A_n)$ is not convergent in the operator norm.

To prove that $A_nf$ is convergent, just notice that $$\|A_n(f)-f\|_{\infty}=\max_{[n/n+1,1]}|f(x)-f(n/n+1)|.$$ By continuity at $x=1$, we know that the latter maximum approaches zero as $n\to\infty$...
• 16.7k
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• 208k
1 vote

### Completely non- normal matrix

The norm of the matrix $A$ is $$A=\begin{bmatrix} a & b \\ 0 & a \end{bmatrix}\implies \| A \| = \frac 12 \left(|b| + \sqrt{|b|^2 + 4|a|^2} \right)$$ (for a proof see Corollary 2 from the ...
• 11.2k
1 vote

• 208k
1 vote

### Prove that the canonical mapping from an infinite dimensional vector space to it's double dual is a one-to-one mapping.

Let $\psi: V \to V^{**}$ be the canonical injection $v\mapsto (g \mapsto g(v))$, we want to prove that $\psi$ is injective. If we have some non-zero $v\in V$ we can extend $\{v\}$ to a basis of $V$ ...
• 111
1 vote
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Your proof is almost correct! The only thing is you need to argue why it is that $Tb_{k+1},\dots,Tb_n$ form a basis for $\text{im } T$. First we will argue that they are linearly independent. Suppose $... • 884 1 vote Accepted ### Prove that$A$is a bounded automorphism of the vector space$X$. Also prove that$\sigma(A) = \{ \lambda \in \mathbb{C} \mid |\lambda| = 1 \}$. When$c=0$,$A$is the identity operator, whose spectrum is just$\{1\}$, so$\sigma(A)=S^1$doesn't hold. We assume$c\not=0$in the next. If$\lambda\in\sigma(A)$,$|\lambda|\le\|A\|=1\$, and we also ...
• 16.8k

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