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21h
comment Limit of a sequence of functions with increasing domains
Perhaps you are fine with a subsequence $f_{2^n}$? Then the domains are increasing and you can consider pointwise convergence on the union of the domains.
21h
comment Limit of a sequence of functions with increasing domains
Ooops, you are right. I have deleted the comment.
22h
answered $C_{c}^{\infty}(\Omega)$ is dense in $L^{\infty}(\Omega)$ with respect to the topology $\sigma(L^{\infty},L^{1})$
22h
comment Show that $\int_{-\infty}^{\infty}f(\xi+i\eta,z_2,\ldots,z_n)e^{i[t_1(\xi+i\eta)+t_2z_2+\cdots+t_nz_n]}d\xi$ is independent of $\eta$
Another way is to differentiate the w.r.t $\eta$ under the integral (justified by Lebesgue's theorem), using the Cauchy-Riemann equation $\partial_\xi +i\partial_\eta$ and calculating the integral of $\partial_\xi$ with the fundamental theorem of calculus.
22h
comment Proving a linear operator is compact: understanding the statement “norm limit of a sequence of finite rank operators”.
This is the definition. Of course "finite rank" is an algebraic condition but operator means continuous linear map between Banach spaces.
22h
comment Riesz representation theorem for $C(\mathbb{T})$
All Borel measures on a compact metric space are regular.
22h
comment Proving a linear operator is compact: understanding the statement “norm limit of a sequence of finite rank operators”.
$\|T\|_{\text{op}}=\sup\lbrace \|T(x)\|_Y: x\in X,\, \|x\|_X\le 1\rbrace$ where $T:X\to Y$.
22h
comment Proving a linear operator is compact: understanding the statement “norm limit of a sequence of finite rank operators”.
Finite rank operators are compact and the subspace of compact operators is closed with respect to the operator norm.
2d
comment Equivalence of two norms.
Don't you know the open mapping theorem (or closed graph or Banach's isomorphy theorem)?
Apr
28
comment A possible norm on a subspace of $C^\infty([0,1])$?
It seems that there are no partitions of unity in this space -- certainly a serious drawback for analytical applications.
Apr
28
answered $\mu(X) \lt \infty$. Then $f_k \to f$ in measure iff for any subsequence $k_l$, there is a subsequence $k_{l_n}$ such that $f_{k_{l_n}}\to f$ a.e.
Apr
26
answered Continuity condition in locally convex F - spaces
Apr
25
comment Convolution of a distribution with a $C^{N}$ function.
This is the standard definition: A distribution $u\in\mathscr D'(\Omega)$ has order $\le n$ if, for every compact set $K\subseteq \Omega$, there is a constant $c\ge 0$ such that $|u(\varphi)|\le c \sup\lbrace |\partial^\alpha \varphi(x)|: x\in\Omega, |\alpha|\le n\rbrace$ for all smooth $\varphi$ with support in $K$.
Apr
22
comment Dual of sum and intersection spaces of Banach spaces.
A good starting point to understand this would be to write down scrupulously what you mean by the intersection (i.e., both spaces are a priory linear subspaces of some vector space), which norm you take there (the maximum), and what you mean precisely by the first equality (the mapping $S: A_0^* \times A_1^* \to (A_0\cap A_1)^*$, $(f,g)\mapsto f|_{A_0\cap A_1} + g|_{A_0\cap A_1}$ is a well-defined linear bijection). Apart from the necessary precision, the main ingredient in the proof is, of course, the Hahn-Banach theorem.
Apr
22
comment Sufficient conditions to have the supremum of a continuous function continuous?
You need compactness of $Y$ to have $f$ uniformly continuous.
Apr
21
comment Sufficient conditions to have the supremum of a continuous function continuous?
The same proof as in math.stackexchange.com/questions/609012/… applies.
Apr
20
revised $r:\mathscr{S'}\rightarrow\mathscr{D'}$, $u\mapsto u|_{\mathscr{D}}$ is not a topological embedding
added 933 characters in body
Apr
20
answered $r:\mathscr{S'}\rightarrow\mathscr{D'}$, $u\mapsto u|_{\mathscr{D}}$ is not a topological embedding
Apr
19
revised $X$ be finite dimensional real NLS , let $x \in X$ , does there exist $T \in \mathcal B(X)$ such that $\{T^n(x):n \in \mathbb N\}$ is dense in $X$?
Corrected spelling.
Apr
19
comment Boundedness of a function on Hilbert space
This is a very old theorem of Hellinger and Töplitz. Nowadays, its a simple consequence of the closed graph theorem.