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2

The set $S$ is convex and symmetric, i.e., $-x \in S$ for all $x\in S$. If $x$ is an interior point of $S$ and $B(x,\varepsilon)\subseteq S$ (where $B(x,\varepsilon)$ is the ball around $x$ with radius $\varepsilon$) you get $B(0,\varepsilon/2)\subseteq S$ since for $\|y\| <\varepsilon/2$ you have $y= \frac 12 (-x) +\frac 12 (x+2y) \in \frac 12 S + \frac ...


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Daniel Fischer's comment says yes, since $\ell^1=c_0^*$. One can give an explicit projection from $(\ell^\infty)^*$ onto $\ell^1$ by saying $$P\Lambda=(\Lambda e_1,\Lambda e_2,,\dots).$$


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The answer to both of your questions is based on the linearity of $f$. For the first question, notice that, if there is even a single $x$ with $f(x)\neq0$, then by multiplying $x$ by a large positive real number $r$, you get $|f(rx)|=r|f(x)|$, which gets arbitrarily large if you take $r$ large enough. So the only way $f$ could be bounded everywhere (rather ...


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To answer your second question: Assume that for some $x_n$ in your sequence it holds that $0 < \vert\vert x_n \vert\vert < 1$ (without loss of generality $0 < \vert\vert x_n \vert\vert$, since otherwise $x_n=0$ and $f(x_n)=0$, which cannot be the case for $n$ large enough). Now define a new sequence $y_n = \frac{x_n}{\vert\vert x_n \vert\vert}$ and ...


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Look at this: http://planetmath.org/banachspacesofinfinitedimensiondonothaveacountablehamelbasis there is refference also. Example is any Banach space of infinite dimension .


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Is $e_0$ supposed to have norm 1? Notice that if we suppose that, we obtain $1=|g_i(e_0)|\leq \Vert g_i\Vert_E\Vert e_0\Vert=\Vert g_i\Vert$, hence $\Vert g_i\Vert_E =1$.


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I think that a direct proof is possible: Let $(u_k)$ be a Cauchy sequence in $\mathcal{B}$. Then, $(u_k)$ is Cauchy in $L^2(0,T; H^1_0)$ and $(\partial_tu_k)$ is Cauchy in $L^2(0,T; H^{-1})$. Since these are Banach spaces, we conclude that there exists $u\in L^2(0,T; H^1_0)$ and $w\in L^2(0,T; H^{-1})$ such that $$\left\{\begin{align}u_k\to ...



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