query: Closed subspace of Banach space
positive_passages.docid: 145952
positive_passages.text: Example of a closed subspace of a Banach space which is not complemented?
negative_passages.docid: 96717
negative_passages.text: Hahn-Banach Theorem for separable spaces without Zorn's Lemma
negative_passages.docid: 43535
negative_passages.text: show that the interval of the form $[0,a)$ or $(a, 1]$ is open set in metric subspace $[0,1]$ but not open in $\mathbb R^1$
negative_passages.docid: 95074
negative_passages.text: Separability of $l^{p}$ spaces
negative_passages.docid: 57609
negative_passages.text: Hahn-Banach proof by extension of basis
negative_passages.docid: 92774
negative_passages.text: Prove that a linear subspace of $C([0,1])$ is closed
negative_passages.docid: 97949
negative_passages.text: Existence of a $\mathbb C$-Banach space isometric to a Hilbert Space but whose norm is not induced by an inner product?
negative_passages.docid: 87655
negative_passages.text: Consequence of the Hahn-Banach Theorem
negative_passages.docid: 88185
negative_passages.text: Proof that a subspace $A$ of a complete metric space $X$ is complete iff $A$ is closed
negative_passages.docid: 152976
negative_passages.text: Real Analysis: open and closed sets
negative_passages.docid: 95513
negative_passages.text: Proving $W$ is a subspace of $P_{2}$
negative_passages.docid: 97621
negative_passages.text: Every separable Banach space is isomorphic to $\ell_1/A$ for some closed $A\subset \ell_1$
negative_passages.docid: 7345
negative_passages.text: Is $C[0,1]$ a Banach space for the $L^1$ norm?
negative_passages.docid: 146146
negative_passages.text: Proof that a closed subset of a compact space is a compact space
negative_passages.docid: 125962
negative_passages.text: The set $A'$ is closed in a $T_1$ space
negative_passages.docid: 109046
negative_passages.text: Is the complement of an open ball in a Banach space connected?
negative_passages.docid: 46326
negative_passages.text: The space $C[a,b]$ with $\|f\|:=\int_a^b |f(x)|d(x)$ is a normed space but not a Banach space