query: $Tf=\sum\limits_{n=1}^\infty f(n)x_n$ is surjective from $\ell^1$ to a separable Banach space
positive_passages.docid: 16220
positive_passages.text: Folland, "Real Analysis", Chapter 5.3, Exercise 36.
negative_passages.docid: 22394
negative_passages.text: Compact operator maps weakly convergent sequences into strongly convergent sequences
negative_passages.docid: 82405
negative_passages.text: For f continuous on $[0,1]$, show that there exist points $\alpha_k$ such that $\sum \limits_{k=1}^n \frac{1}{f'(\alpha_k)} = n $
negative_passages.docid: 7345
negative_passages.text: Is $C[0,1]$ a Banach space for the $L^1$ norm?
negative_passages.docid: 146306
negative_passages.text: A continuous surjective function from $(0,1]$ onto $[0,1]$
negative_passages.docid: 83040
negative_passages.text: Proving that the sequence $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$ is boundedly convergent on $\mathbb{R}$
negative_passages.docid: 136321
negative_passages.text: Show that if $\sup\big\{\sum\lvert\, f(a)\rvert\big\}  0\}$ is countable.
negative_passages.docid: 125623
negative_passages.text: Subspaces of separable normed spaces
negative_passages.docid: 84818
negative_passages.text: Convergence of $\sum_{n=1}^{\infty}x_n\overline{y_n}$ for every $(x_n)\in c_0$ implies $\sum_{n=1}^{\infty}|y_n|<\infty$
negative_passages.docid: 7069
negative_passages.text: On the product of $\mathfrak c$-many separable spaces
negative_passages.docid: 134859
negative_passages.text: Let $(X, \mathcal{T})$ be a metrizable space. Prove that $X$ Lindelöf implies that $X$ is separable.
negative_passages.docid: 21391
negative_passages.text: Proving Functions are Surjective
negative_passages.docid: 63285
negative_passages.text: Prove that this space is not Banach
negative_passages.docid: 132491
negative_passages.text: How to show that $C=C[0,1]$ is a Banach space
negative_passages.docid: 153121
negative_passages.text: show that $\{x_n\}$ is a cauchy sequence
negative_passages.docid: 131927
negative_passages.text: lim inf $|a_n|=0 \implies \sum_{k=1}^\infty a_{n_k}$ converges
negative_passages.docid: 44706
negative_passages.text: Behavior of $\sum a_n x^n$ given $\sum |a_n - a_{n-1}| < \infty$