query: Lemma 3.3-7 and Theorem 3.6-2 in Kreyszig's "Introductory Functional Analysis With Applications": What if completeness is lost?
positive_passages.docid: 21459
positive_passages.text: Orthogonal complement examples
negative_passages.docid: 86717
negative_passages.text: Completeness of $\langle \mathscr{C} [0, 1], \| \cdot \|_1 \rangle$
negative_passages.docid: 90435
negative_passages.text: Prove that there exist linear functionals $L_1, L_2$ on $X$
negative_passages.docid: 131664
negative_passages.text: Why is any Lebesgue-Stieltjes measure on $(\mathbb{R},\mathcal{B}_\mathbb{R})$ not complete?
negative_passages.docid: 24478
negative_passages.text: What is the name of this theorem, and are there any caveats?
negative_passages.docid: 120170
negative_passages.text: A late-diverging "approximating solution" for a system of functional equations
negative_passages.docid: 50137
negative_passages.text: Using Cantor's lemma in a proof of convergence
negative_passages.docid: 118617
negative_passages.text: Completeness and Incompleteness
negative_passages.docid: 44457
negative_passages.text: Why we define the completeness of a space by the converge of a Cauchy sequence rather than a normal sequence?
negative_passages.docid: 66951
negative_passages.text: Proving the existence of a sequence of functions converging to 0 with an added property
negative_passages.docid: 139540
negative_passages.text: Real Analysis, Folland Problem 1.3.15 Measures
negative_passages.docid: 68255
negative_passages.text: Theorems in complex analysis.
negative_passages.docid: 35115
negative_passages.text: Questions on how to prove that a set of connectives is NOT functionally complete
negative_passages.docid: 35751
negative_passages.text: Is There a Problem with This Professor's Proof Concerning Interior and Boundary Points?
negative_passages.docid: 152590
negative_passages.text: At most finitely many (Hamel) coordinate functionals are continuous - different proof
negative_passages.docid: 46211
negative_passages.text: $(K[x,y,z]/(x^2+y^3+z^7))_{(x,y,z)}$ is a UFD.
negative_passages.docid: 79011
negative_passages.text: If $(F:E)<\infty$, is it always true that $\operatorname{Aut}(F/E)\leq(F:E)?$