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Let $E$ be a separable Banach space and $(y_j)_{j\in \mathbb N}$ be a sequence in $B(0,1)$ which is dense in $B(0,1)$. Where $B(0,1)$ is the open ball of center $0$ and radius $1$. Show that the map \begin{align} A:~&l_1~~~~~~~~~\longrightarrow E\\ &(x_i)_{i\in\mathbb N}\longmapsto \sum_{i=1}^\infty x_iy_i \end{align} is linear, continuous, surjective.

I don't have any idea to show the function $A$ is surjective. Can someone please help me.

Quyle
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  • A version of the open mappin theorem (like, e.g., in Rudin's bok) tells you that it is enough to show that the closure of the image of the unit ball contains a ball in $E$. This condition follows from the density of $(y_j)_{j\in\mathbb N}$. – Jochen Feb 20 '17 at 11:29

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Note $A$ is continuous: $$\|A(x)\|=\left\|A\left(\sum_{n\in\Bbb N} x_n e_n\right)\right\| = \left\|\sum_n x_n y_n\right\|≤\sum_i|x_i|\,\|y_i\|≤\sum_i|x_i|=\|x\|$$ so its even contractive.

Let $y^*$ be in the unit ball of $E$, then there exists an $y_{n_1}$ with $\|y^*-y_{n_1}\|≤1/2$. It follows $2(y^*-y_{n_0})$ is in the unit ball of $E$ and you have a $y_{n_1}$ with $\|2(y^*-y_{n_0})-y_{n_1}\|≤1/2$, in other words $\|y^*-y_{n_0}-\frac{y_{n_1}}{2}\|≤2^{-2}$. Carry this out inductively to get a sequence $y_{n_k}$ so that $$\left\|y^*-\sum_{l=0}^k2^{-l}y_{n_l}\right\|≤2^{-k-1}$$ So $$A\left(\sum_{l=0}^\infty 2^{-l}e_{n_l}\right)=\sum_{l=0}^\infty 2^{-l}y_{n_l}=y^*$$ and the image of $A$ contains the unit ball. Since the unit ball is adsorbing it follows $A$ is surjective.

Remark: this proof can be adapted to show that if $A:X\to Y$ is a continuous map between Banach spaces so that $A(B_1(0)_X)$ is dense in $B_1(0)_Y$ that the map must be a surjection.

s.harp
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