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May
22
comment function defined as integral of borel function
In Fubini's formula for $\int_A\int_B f(x,y) d\nu(y) d\mu(x)$ one needs and proves the measurability of $x\mapsto \int_B f(x,y)d\nu(y)$. This is very similar to your measurability question, isn't it?.
May
21
comment function defined as integral of borel function
Look at (the proof of) Fubini's theorem. Measurability of the integral w.r.t. one of the variables is part of the statement.
May
20
comment Showing bounded linear operator has closed image
Your proof is correct.
May
19
answered Hölder continuous functions are of 1st category in $C[0,1]$
May
19
comment Hölder continuous functions are of 1st category in $C[0,1]$
Every continuous function on a compact space is uniformly continuous.
May
15
comment Compute Quotient Space
Look at $F:c \to c_0$, $(x_n)_n \mapsto (x_n - x_\infty)_n$ where $x_\infty=\lim x_n$. $F$ vanishes on $Y$ and thus induces a well-defined map $c/Y\to c_0$ which is easily seen to be a bijection.
May
14
comment Compute Quotient Space
$S$ isn't closed in $c_0$. It is a dense subspace.
Apr
27
answered Is the following set of infinite absolutely convex combinations closed?
Apr
15
awarded  Custodian
Apr
15
comment Schauder basis for $c_0$
Well, the sequence $(e_k)_{k\in\mathbb N}$ is then a matrix.
Apr
15
reviewed Approve Show that the Euclidean algorithm works for Gaussian integers.
Apr
15
comment Schauder basis for $c_0$
Perhaps, the confusion comes from your somewhat unprecise writing $e_k=\delta_{j,k}$ which should be $e_k=(\delta_{j,k})_{j\in \mathbb N}$.
Mar
18
answered For any sequence from Frechet spaces there exists a sequence that takes it to zero
Mar
18
comment properties of a Köthe space s
What do you mean be $\log^\beta n$? Is it $(\log(n))^\beta$?
Mar
18
answered properties of a Köthe space s
Mar
18
comment Weak convergence + compactness = strong convergence?
A compact space does not a have a strictly weaker Hausdorff topology.
Mar
17
comment Characterization of product of dual dual Hilbert spaces.
Define $F_1(x_1)=F(x_1,0)$ and $F_2(x_2)=F(0,x_2)$.
Mar
12
answered Completion of a vector space inside a given Banach space
Feb
26
comment Suppose that $ T$ is a linear operator from a normed space $ X $ into normed space $ Y $.Prove that the following are equivalent
You could add iii: $T(K)$ is bounded for every compact subset of $X$. Since weakly compact sets are bounded, ii implies iii which gives i because you only have to prove that images of null sequences are bounded.
Feb
25
comment About the weak* closedness of the kernel of a continuous linear functional
Another idea is to use the real case and write $\varphi(x)= \Re \varphi(x) + i \Im \varphi(x) = \Re \varphi(x) - i \Re(i \varphi(x)) = \Re \varphi(x) - i \Re \varphi(ix)$