# Tagged Questions

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### Show that $B(X)$ is semisimple for a Banach space $X$ [duplicate]

Show that $B(X)$ is a semisimple Banach algebra, where $X$ is a Banach space. That is, to show that rad $B(X)=\{0\}$, or equivalently, to show $\sigma(AT)={0} \, \forall T\in B(X)\Rightarrow A=0$. I ...
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### Banach Alaoglu different proofs

While trying to prove Banach-Alaoglu theorem I noticed the differrent equivalent definitions of compactness. When I tried to find a proof of Banach-Alaoglu I found a proof in Pedersen Analysis Now and ...
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### closed unit ball in a Banach space is closed in the weak topology

Let $V$ be a Banach space. Show that the closed unit ball in $V$ is also closed in the weak topology. I know this is a consequence of the statement any closed convex subset in $V$ is closed in the ...
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### Compact embedding

Prove that the embedding $j\colon (C^1[0,1],\|\cdot\|)\to(L^1[0,1],\|\cdot\|_{L^1})$ where $\|f\|=\max\{\|f\|_\infty,\|f'\|_\infty\}$ and $\|f\|_\infty$ denotes the supremum norm, ...