query: The image of bounded linear operator is closed iff the image of its subspace is closed
positive_passages.docid: 157303
positive_passages.text: An operator has closed range if and only if the image of some closed subspace of finite codimension is closed.
negative_passages.docid: 136872
negative_passages.text: Why uniform closure $\mathscr{B}$ of an algebra $\mathscr{A}$ of bounded complex functions is uniformly closed?
negative_passages.docid: 61931
negative_passages.text: Positive operator is bounded
negative_passages.docid: 145952
negative_passages.text: Example of a closed subspace of a Banach space which is not complemented?
negative_passages.docid: 83970
negative_passages.text: Proof of an elementary property of Projection Operators
negative_passages.docid: 24050
negative_passages.text: Characterization of the Closure of a Set
negative_passages.docid: 143674
negative_passages.text: Space of bounded functions is complete
negative_passages.docid: 28685
negative_passages.text: If every linear operator on $X$ is bounded then is $\dim X<\infty$?
negative_passages.docid: 108740
negative_passages.text: $C^1[a,b]$ is closed in $C[a,b]$
negative_passages.docid: 140756
negative_passages.text: Image of closed unit ball under a compact operator
negative_passages.docid: 115272
negative_passages.text: Showing that projections $\mathbb{R}^2 \to \mathbb{R}$ are not closed
negative_passages.docid: 2129
negative_passages.text: Showing that the derivative operator is not bounded on $L^2 (\mathbb{R})$
negative_passages.docid: 8414
negative_passages.text: Proof verification: Set of partial limits of a sequence is closed
negative_passages.docid: 12490
negative_passages.text: A map is continuous if and only if for every set, the image of closure is contained in the closure of image
negative_passages.docid: 11558
negative_passages.text: How to show that the limit of compact operators in the operator norm topology is compact
negative_passages.docid: 80618
negative_passages.text: Surjective bounded operator in Banach spaces without bounded right-inverse
negative_passages.docid: 66531
negative_passages.text: $\|h(x)-h(y)\|\geq 3\|x-y\|$ prove that image of open is open