13 Why does integral and the imaginary part commute? 11 Prove local minimum of a convex function is a global minumum (using only convexity) 9 A real function which is additive but not homogenous 9 If $f,g$ are continuous and $g$ is $1$-periodic, $\int_0^1 f(x)g(nx)dx \xrightarrow[n\to\infty]{} \int_0^1 f \int_0^1 g$ 9 Subset $A$ of $\mathbb{C}$ such that $\prod_{a\in A}(1+a)=1$

### Reputation (4,178)

 +35 Example of use of closed graph theorem over open mapping theorem and bounded inverse theorem +45 In a metric space, is every open subset a non-redundant union of open balls? +10 Mistake in proof of $\overline{C_c(X)} = C_0(X)$.. +10 How can I show the statement: “every Cauchy sequence converges” can replace the completeness axiom?

### Questions (18)

 14 Can $\sum_{k\in M}\frac{1}{k}$ be a large integer? 6 Mistake in proof of $\overline{C_c(X)} = C_0(X)$.. 5 Can weak convergence be checked on an orthonormal basis? 4 Ultraproduct of partitions of an $L^p$-space isomorphic to itself? 3 Convention: A degenerate cuboid is a rectangle; is its surface area twice the rectangle's area or only once?

### Tags (96)

 54 calculus × 15 25 linear-algebra × 7 51 functional-analysis × 36 24 complex-numbers × 4 47 real-analysis × 24 23 complex-analysis × 16 26 integration × 6 20 convex-analysis × 3 25 limits × 9 18 sequences-and-series × 9

### Bookmarks (5)

 18 Is the closedness of the image of a Fredholm operator implied by the finiteness of the codimension of its image? 11 Can every closed subspace be realized as kernel of a bounded linear operator from a Banach space to itself? 4 $\varphi(A(x))$ continuous $\Rightarrow$ $A$ continuous? 2 Prove continuity of a bounded linear operator 2 Existance of element in Banach subspace which norm is equal to 1 and to distance between this element and the other subaspace.