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comment $T:X \to Y$ bounded linear map and $X$ separable implies $Y$ is separable?
Concerning reflexivity you can also argue with the weak* compactness of the closed unit balls. If $T:X\to Y$ is onto then the unit ball of $Y$ is contained in a multiple of $T(B_X)$ (by the open mapping theorem) which is weakly compact. Hence $B_Y$ is relatively weakly compact and weakly closed, hence weakly compact.
Jul
2
comment Is it possible to approximate $cos(x)$ with a linear combination of Gaussians $e^{-x^2}$?
Look at "convolution with Gaussian kernels". This, by the way, is the main trick how Weierstrass proved his famous approximation theorem.
Jun
29
comment Functional Analysis (Topological and Isometric Isomorphisms)
I think the OP meant linearly isomorphic. However, David's example works, of course. To see that $c$ and $c_0$ are not isometric one could observe that the unit ball of $c$ has extreme points but that of $c_0$ does not.
Jun
11
answered Matrix-like Representation of any linear map using Hamel Basis
Jun
5
awarded  Popular Question
Jun
5
comment No surjective bounded linear map from $\ell^2(\mathbf{N})$ to $\ell^1(\mathbf{N})$
A similar argument. $\ell^2$ is reflexive, quotients of reflexive spaces are reflexive and $\ell^1$ is not reflexive.
Jun
3
accepted Compact subsets of the plane with connected complement
Jun
3
comment Compact subsets of the plane with connected complement
@MikeMiller If I understand your comment well this answers the question. If you formulate it as an answer I would like to accept it.
Jun
2
asked Compact subsets of the plane with connected complement
May
31
comment Show that the sequence $\sum _{k=1}^{\infty}\frac{e^k}{k\log(k+1)}$ is convergent in $c_0$,but not absolutely convergent in $c_0$
Is $e^k=\exp(k)$ or do you mean perhaps the sequence $e^k=(\delta_{n,k})_{n\in\mathbb N}$?
May
30
comment If $x \in c_0 $ and $\sum_{i=1}^{\infty}|x_n|< \infty $ $ \implies x \in l^p$ for $p \in (1,\infty)$
$\sum_{n=1}^\infty |x_n| <\infty$ implies $x\in c_0$.
May
30
comment Folland, “Real Analysis”, Chapter 5.3, Exercise 36.
For the open unit balls you would need the $\epsilon$, for the closed balls it works without.
May
27
comment Why $p\{N>n\}=p\{X_1+…+X_n\leq x\}$.
The sets are equal. It might help to call $S_n(\omega)=X_1(\omega)+\cdots X_(\omega)$. You only need that $(S_n(\omega))_n$ is increasing for each $\omega$.
May
26
answered Folland, “Real Analysis”, Chapter 5.3, Exercise 36.
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.