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Let $X$ be a compact Hausdorff space. A continuous map $f:X \to X$ defines a bounded linear operator $T_{f}$ on the Banach space $C(X)=\{\phi:X\to \mathbb{C} \mid \phi\; \text{is continuous}\}$ with …

Assume that $X$ and $Y$ are two Banach spaces and $T:X\to Y$ is a bounded surjective linear operator. A consequence of the Michael selection theorem is that:"There is a continuous function $g:Y\to X$ …

Assume That $A$ is a commutative complex Banach algebra. Let $G$ be the connected component of invertible elements containing the identity. Is there an smooth embedded curve $c:(-\epsilon, \epsil …

Assume that $V$ is a finite dimensional real or complex normed linear space. Let $Iso(V)\subset GL(V)\subset L(V)$ be the space of linear isometric endomorphisms, invertible endomorphism and li …

Edit: According to comments of Eric Wofsy and Yemon Choi I edit the question. For a (compact) topological space $X$, we put $A=\{f:X\to \mathbb{C}\mid f\text{ is bounded}\}$. We define a semi-n …

Inspired by Pitt's theorem and this post we ask the following question: First we remind the Pitt's theorem and also introduce a particular Banach space as averaging of $\ell^p$ spaces for $1\leq p …

Let $F\subset B(H)$ be a finite dimensional subvector space of the space of all bounded operators on a Hilbert space. Question: Is there an upper bound for $$\{|tr(T)| \text{where} \quad T\in …

Let $A$ be a Banach algebra. Is there a Banach algebra $B$ which contains $A$ but the spectrum of each elements of $B$ has empty interior(as a subset of $\mathbb{C}$)? The motivation comes from the …

A Banach space $X$ is called Hilbert-irreducible if it satisfies the following condition: If a subspace $Y\subset X$ satisfies the parallelogram equality, then $Y$ is necessarilly a one dimensi …

The space of Schwartz functions on the plane is denoted by $\mathcal{S}$. The usual multiplication and the convolution multiplication on $\mathcal{S}$ are denoted by $m_1$ and $m_2$, respectively. Th …

To what extent have all unital $C^*$ algebras $A$ with the following property been classified? Is there a simple $C^*$ algebra with this property? Does $C(K)$ satisfy this property, where $K$ is an ar …

Regarding the non commutative case, I just realized that a non commutative $C^*$ algebra has a non scalar element $1+a$ with connected spectrum $\{1\}$ where $a$ is a non zero nilpotent eleme …

In this question the norm of $L^{P}[0,1]$ is denoted by $\parallel . \parallel _{p}$. Let $p$ and $q$ be two arbitrary real numbers with $2<p<q$. Assume that $S$ is a subvector space of $L^ …

Assume that $X$ is a Banach space. Is there a continuous map $f:X\to X$ such that $f$ is nowhere Frechet differentiable, but its restriction to every finite dimensional subspace is every where Frechet …

Assume that $X$ and $Y$ are Banach spaces and $T:X\to Y$ is a bounded surjective linear map. Is there a Gateaux differentiable function $g:Y\to X$ such that $T\circ g=Id_{Y}$?