query: In a Banach space X, if $\forall f \in C(M)$, $f$ is bounded, show M is compact
positive_passages.docid: 126759
positive_passages.text: Unbounded continuous function on non-compact metric space
negative_passages.docid: 144400
negative_passages.text: Are compact subsets of metric spaces closed and bounded?
negative_passages.docid: 19294
negative_passages.text: ‎If ‎‎$‎X$ is an infinite-dimensional Banach space and ‎‎$‎‎u‎\in ‎B(X)‎$ ,then $\bigcap_{v\in K(X)}\sigma(u+v) =\cdots$
negative_passages.docid: 14509
negative_passages.text: Contraction Map on Compact Normed Space has a Fixed Point
negative_passages.docid: 137067
negative_passages.text: For Banach space there is a compact topological space so that the Banach space is isometrically isomorphic with a closed subspace of $C(X)$.
negative_passages.docid: 102048
negative_passages.text: Closure of a compact set in Regular space $ X $
negative_passages.docid: 71011
negative_passages.text: Criteria of compactness of an operator
negative_passages.docid: 28391
negative_passages.text: Prove the compactness of a set B
negative_passages.docid: 104607
negative_passages.text: Hahn-Banach theorem, dual space
negative_passages.docid: 7081
negative_passages.text: isometry $f:X\to X$ is onto if $X$ is compact
negative_passages.docid: 136583
negative_passages.text: Infimum is a continuous function, compact set
negative_passages.docid: 36149
negative_passages.text: A uniformly continuous function maps bounded set to bounded sets
negative_passages.docid: 138607
negative_passages.text: Proof that a limit point compact metric space is compact.
negative_passages.docid: 65358
negative_passages.text: Prove $X$ is compact iff for each family $\{ F_\alpha\}_{\alpha\in I}$ of closed subsets of $X$ with the FIP, we have $\cap F_\alpha\neq\varnothing$.
negative_passages.docid: 67259
negative_passages.text: Can we prove that a bounded closed subset of $\mathbb R^n$ is compact without Axiom of Choice?
negative_passages.docid: 11114
negative_passages.text: Is a closed subset of a compact set (which is a subset of a metric space $M$) compact?
negative_passages.docid: 113151
negative_passages.text: Does $\sum\limits_{i=1}^{\infty}|a_i||x_i| < \infty$ whenever $\sum\limits_{i=1}^{\infty} |x_i| < \infty $ imply $(a_i)$ is bounded?