8
votes
Accepted
Semi-continuity in quasi-finite morphisms without properness
The function is lower semi-continuous, see EGA4, Prop. 15.5.1.
Note that since you assume the fibres reduced hence (because the characteristic is 0) geometrically reduced, the map $f$ is in fact étale;...
- 4,492
6
votes
Continuity concepts for correspondences
Yes, these are the so-called Vietoris topologies. The upper Vietoris topology has a subbase consisting sets of the form $\{F\in 2^Y\mid F\subseteq O\}$ with $O$ open and the lower Vietoris topology ...
6
votes
Geometric applications of Ekeland's variational principle
Here is a quick application of the Ekeland's Variational Principle to Spectral Theory. Let $A$ be a bounded linear symmetric operator on a Hilbert space $H$, and let $\mathbb{S}$ be the unit sphere ...
- 60.5k
6
votes
A net of lower semicontinuous functions
Take the subgraphs
$$
U_\alpha = \{(x,y) \in [0,1]\times\mathbb R \mid y<f_\alpha(x)\}
$$
which are open sets. The result in question would be a consequence of:
Let $X$ be a separable metric ...
- 41.1k
3
votes
A net of lower semicontinuous functions
Let us denote $I=[0,1]$ and let us choose $\varepsilon_n=2^{-n}$.
Notice that in the situation in the question we have $f=\sup\limits_{\alpha\in A} f_\alpha$, and thus $f$ is a lsc function.$\...
- 4,713
3
votes
Accepted
Is a convex, lower semicontinuous function that is bounded from below, actually continuous?
Though not entirely in the same setting, as can be seen from these lecture notes my reasoning seems to hold. In the lecture notes, one considers barrelled spaces but the local boundedness at some ...
- 651
3
votes
Accepted
Takahashi minimization theorem for lower pseudo-continuous functions on complete metric spaces
The Takahashi Theorem holds also for lower pseudo-continuous functions.
To derive a contradiction, assume that $f:X\to [0,+\infty]$ a proper lower pseudo-continuous function such that for any point $...
- 41.8k
2
votes
Accepted
Weak lower semicontinuity of functional with two arguments
This functional is sequentially weakly lower semicontinuous under fairly mild assumptions on $f$.
We need that $f$ is non-negative, continuous and bounded from above.
Let $u_n \rightharpoonup u$ in $...
- 1,714
2
votes
Rank of a linear combination of linear operators
A negative answer to the first question is given by
$$ X=\left[ \begin{array}{ccc} 0 & 1 & 0\\
0 & 0 & 0\\ 0 & 0 & 0 \end{array} \right],\ \
Y=\left[ \begin{array}{...
- 50.8k
2
votes
Accepted
Lower semi-continuity of length-dependent functional
Let $A_n := \{x \in \ell^1 \colon x_1 \not= 0, \ldots, x_{n-1} \not= 0\}$, $g_n \colon \ell^1 \to [0,\infty]$ be defined by $g_n(x) := f(x_n)$ if $x \in A_n$ and $g_n(x) := 0$ if $x \not\in A_n$. Then ...
- 2,101
2
votes
Accepted
Right-continuity of covering number
Without additional assumptions on the metric space, it may appear that for every $\varepsilon>1$ the covering number equals 1, but for $\varepsilon=1$ it is infinite. For example, let positive ...
- 109k
2
votes
Accepted
A question about a realcompact space and upper semicontinuous function
The following characterisation is well known. It can be found in Engelking's book as Theorem 3.11.10.
Theorem: A Tychonoff space $X$ is realcompact if and only if for each $p\in\beta X\setminus X$ ...
- 5,596
1
vote
Problem in understanding maximum principle for subharmonic functions
While this particular case is easier (as shown by Arnab), it also follows from the very useful, and not all that well known Katětov–Tong insertion theorem stating that:
If $X$ is a normal topological ...
- 151
1
vote
Problem in understanding maximum principle for subharmonic functions
Proposition $:$ If $f$ is a u.s.c. function on $\Omega$ and bounded above, then there is a sequence $f_1 \geq f_2 \geq \cdots$ of continuous functions on $\Omega$ that are bounded above and that ...
1
vote
Accepted
On the additive property of the subdifferential of lower semicontinuous functions
In part (P3) of Definition 2.1 in the paper you linked, it is also required that $g$ be $\partial$-differentiable at $x$ (meaning that both $\partial g(x)$ and $\partial(-g)(x)$ are nonempty), which ...
- 128k
1
vote
Accepted
Weak lower semicontinuity of a sequence of Riemann sums
No. E.g., for natural $K\ge2$, let
$$f^K:=1-g^K,$$
where, for $t\in[0,1]$,
$$g^K(t):=\sum_{j=0}^{K-1}\Big(1-K^2\Big|t-\frac{j+1/2}K\Big|\Big)_+$$
and $u_+:=\max(0,u)$. That is, for each $t\in[0,1]$, ...
- 128k
1
vote
Is a convex, lower semicontinuous function that is bounded from below, actually continuous?
it is true that a convex function is continuous on the relative interior of it's domain, but not necessarily on the relative boundary, consider the function $f(x)=0$ if $0<x<1$ and $f(x)=1$ if $...
- 11
1
vote
Looking for a reference: $f$-divergences are lower semicontinuous
I you further assume that $f$ is lower semicontinuous then the $f$ divergence is weakly lsc w.r.t. to both its primary argument $P$ and reference measure $Q$, i-e
$$
D_f(P\| Q)\leq \liminf\limits_{n\...
- 5,380
1
vote
Looking for a reference: $f$-divergences are lower semicontinuous
This can be seen analogous as for the KL divergence using a duality representation. For the case, where $f'(\infty)=\infty$, i.e. $f$ has super linear growth, we can define thanks to the convexity of $...
- 1,081
1
vote
Accepted
Lower semi-continuity of induced function on sequences
Let $x = (x_n) \in \ell^p(X)$ and $F_N(x) := \sum_{n=1}^N f(x_n)$, $N \in \mathbb{N}$. First if each $F_N$ is l.s.c. (weakly or not), then $F = \sup_{N \in \mathbb{N}} F_N$ is l.s.c., second if each $...
- 2,101
1
vote
Function series of normal lower semi-continuous functions
Since $\left\{ x\in X:f\left( x\right) <\lambda \right\} =\left\{
\begin{array}{cl}
\emptyset, & \lambda\le 0, \\
X, & \lambda>1 \\
\overline{U}_k, & \frac 1{2^k}<\lambda\le \...
- 1,367
1
vote
Accepted
Rank of a linear combination of linear operators
I think one can cook a similar counter-example to that you mention. Consider the matrices: $X = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ ...
- 7,300
1
vote
Regarding upper semicontinuity of a function
Fix $A$ and take a sequence $A_k\to A$, denote $\mu_E(A_k)=a_k$. We have to prove that $\mu_E(A)\geqslant \limsup a_k$. Assume the contrary. Then
$a:=\limsup a_k>\mu_E(A)\geqslant 0$. Passing to a ...
- 109k
1
vote
Accepted
Goldowsky-Tonelli theorem for upper semi continuous function
Let's assume that $\partial^{e} f(. )$ denotes the right derivative (the left derivative can be handled similarly).
Claim: The function $g(t)=f(t)-t\partial^{e} f(t)$ is indeed (weakly) decreasing ...
- 14.2k
1
vote
When do convexity and lower semicontinuity imply continuity?
Let me resolve a special case of the question, by leveraging the examples mentioned above. Suppose $(K,\rho)$ is a compact metric space and $X$ is the set of Borel probability measures on $K$ endowed ...
- 537
1
vote
Accepted
If a function follows another one's range order, can we say it follows some continuity properties?
No, let $X = \{0\} \cup \{\frac{1}{n}: n \in \mathbb{N}\}$, define $f(t) = -t$ for all $t \in X$, and define $g(t) = \begin{cases}0&t > 0\cr 1&t = 0\end{cases}$.
- 42.8k
1
vote
properties of orderd upper and lower semi continuous functions
In general there is no continuous function between u.s.c. $f$ and l.s.c. $g\leq f$. For example, take $f(x)=1, 0\leq x\leq 1;\; f(x)=0, 1<x\leq 2$, this is u.s.c.
Now $g(x)=1, 0\leq x<1;\; g(x)=...
- 91.8k
Only top scored, non community-wiki answers of a minimum length are eligible