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        Questions tagged [banach-spaces]

        A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

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        votes
        1answer
        82 views

        Complemented subspaces of Lorentz sequence spaces?

        Let $d(\textbf{w},p)$, $1\leq p<\infty$, denote the Lorentz sequence space, where $\textbf{w}=(w_n)_{n=1}^\infty\in c_0\setminus\ell_1$ is a normalized decreasing weight. Is there very much known ...
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        votes
        0answers
        64 views

        Dual Lorentz spaces

        MO seems the perfect place to ask for the following question. Denote the Lorentz spaces on an arbitrary measure space $(E,\mu)$ by $L^{p,q}=L^{p,q}(E,\mu)$, and by $p'$ the conjugate index of $p$. ...
        5
        votes
        1answer
        190 views

        Quasinilpotent , non-compact operators

        If $X$ is a separable Banach space and $(\epsilon_n)\downarrow 0$, can we find a quasinilpotent, non-compact operator on $X$ such that $||T^n||^{1/n}<\epsilon_n$ for all $n$? I suspect the answer ...
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        0answers
        68 views

        Duality mapping of a the space of continuous functions? [closed]

        The duality map $J$ from a Banach space $Y$ to its dual $Y^{*}$ is the multi-valued operator defined by: $J(y)=\{\phi\in Y^{*}:\, \left< y,\phi\right >=\Vert y\Vert^{2}=\Vert \phi \Vert^{2}\},...
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        votes
        0answers
        145 views
        +100

        Approximation of a compactly supported function by Gaussians

        Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function whose support is a closed interval, e.g. $\text{supp}(f)=[a,b]$. Then $f$ can be approximated (e.g. in $L^2$) by a linear combination of Gaussian ...
        3
        votes
        2answers
        111 views

        Reference request: $\alpha$-Hölder spaces as double duals

        If $(X,d)$ is a complete metric space, we define the $\alpha$-H?lder class $\Lambda_\alpha(X)$ as the subset of $C_b(X)$ satisfying that $$ \sup_{x \neq y} \frac{|f(x) - f(y)|}{|x - y|^\alpha}. $$ ...
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        votes
        1answer
        232 views

        Peak sets and Choquet boundary of a function algebra

        I have two problems to ask. Let $A$ be a function algebra of $C(K)$. $t\in K$ is said to be a peak point of $A$ if $\exists~f\in A$ s.t. $|f(t)|=\|f\|$ and $|f(s)|<|f(t)|$ for any $s\neq t$. ...
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        vote
        0answers
        54 views

        Regularity of superposition operator generated by function between Banach spaces

        Let $E$, $F$ be Banach spaces, $D$ be open in $E$, and $K=[0,1]$. Given $\varphi\colon K\times D\to F$ I call $$ \varphi^\sharp\colon D^K\to F^K,\quad u\mapsto \varphi(\cdot,u(\cdot)) $$ the ...
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        vote
        0answers
        36 views

        Monotonicity of the norms on the sequence spaces 2

        This is a complement of my previous question about the sequence spaces (I'm afraid, there will be a third part). Let $\rho:[0,+\infty)^{\mathbb{N}}\to[0,+\infty] $ satisfy the following properties: $...
        5
        votes
        1answer
        135 views

        Hahn-Banach smoothness of $Y^{**}$ in $X^{**}$

        A subspace $Y$ of a Banach space is said to be Hahn-Banach smooth if every $f\in Y^*$ has unique norm preserving extension to whole $X$. This notion is related to many other geometric properties of ...
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        votes
        1answer
        69 views

        Vectors concentrated on one coordinate

        Suppose $X$ is a Banach space, $(e_i)$ a normalized basis, $(e_i^*)$ the biorthogonal functionals, and $Y$ a finite codimensional subspace of $X$. Given $N$ and $\varepsilon$, can we find $x\in Y$ ...
        2
        votes
        1answer
        157 views

        Does the norm on a sequence space have to be monotone?

        Let $\rho:[0,+\infty)^{\mathbb{N}}\to[0,+\infty] $ satisfy the following properties: $\rho(\lambda u)=\lambda\rho( u)$, for every $u\in [0,+\infty)^{\mathbb{N}}$ and $\lambda\ge 0$; $\rho(u+v)\le \...
        2
        votes
        1answer
        132 views

        Can a bijection between function spaces be continuous if the space's domains are different?

        It is well-known that any bijection $\mathbb{R} \rightarrow \mathbb{R}^2$ cannot be continuous. But suppose we have the two spaces $A = \{f(x):\mathbb{R^2}\rightarrow \mathbb{R} \}$ and $B = \{f(x):\...
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        votes
        0answers
        41 views

        Show that $\Phi_{P(X)}=\hat{X}$

        Let $X$ be a compact subset of $\mathbb {C^n}$. The polynomial convex hall of $X$ is the set $\hat{X}=\{z\in \mathbb {C^n}: \left|P(z) \right|\leq \left||P |\right|_\infty , \text{for all ...
        0
        votes
        1answer
        169 views

        Regarding orthogonality in Banach space

        Let $(X,\|.\|)$ be a Banach space over the real line. Let $x\neq 0$ and $y\neq 0$ be in $X$, then $x$ is said to be orthogonal to $y$ if $\|x+\lambda y\|\geq\|x\|$ for every real number $\lambda$. ...

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