norms – Embedding of a Banach space into a Hilbert space

Let $mathbb H$ be a Hilbert space and let $mathbb B$ be a Banach space continuously embedded in $mathbb H$ and distinct from $mathbb H$. Is it true in general that $mathbb B$ is an $F_sigma$ of $mathbb H$? I know several examples where it is actually true, but the proof seems to depend on the particular structure of the norms on $mathbb B, mathbb H$, so I wonder if there is a general result of the sort.

Different definitions of equivalent norms

I’m trying to show that the following definitions are equivalent

Two norms over a field F are equivalent if:

  1. there exists two costants A,B such that A|x|_1<|x|_2<B|x|_1 for every x in F
  2. there exists c in Re such that |x|_1^c = |x|_2

convex optimization – Reference request: second-order def of strong convexity wrt general norms

A function $f: R^n rightarrow R$ is said to be C-strongly convex with respect to a norm $|cdot|$ if for all $x,y$ and $lambda in (0,1)$ $$f(lambda x + (1-lambda)) le lambda f(x) + (1-lambda) f(y) – C lambda (1-lambda) |x-y|^2.$$

When $f$ is twice differentiable it seems that an equivalent condition for C-strong convexity is that all second derivatives in directions $u$ are at least $approx C |u|^2$ (maybe there is a constant missing), namely for all $x,u$ $$D^2_xf(u,u) gtrsim C |u|^2,$$ where $D^2_x f(u,u) = g”(0)$ for $g(t) = f(x + tu)$.

Does anyone have a reference for this fact? When the norm is Euclidean this equivalence is in basically every book on convex analysis and optimization, but for the life of me I cannot find a reference for general norms $|cdot|$.

Such fact has been used in its dual form (for strong smoothness) in several places in probability, optimization and online learning, for example this paper of Pinelis, this paper by Juditsky and Nemirovski, and this paper by Kakade et al., but without a reference (maybe it is obvious for them but my current proof is messy and has to go through mid-point strong convexity, etc.).

mathematical optimization – SemidefiniteProgramming for operator norms: Stuck at the edge of dual feasibility

I’m trying to calculate operator norms of linear transformations over space of matrices. For instance, find norm of $f(A)=XA$ by optimizing following:

$$max_{|A|=1} |XA|$$

This looks like a semidefinite programme, but I’m having trouble solving it with SemidefiniteOptimization. Simplest failing example is to find operator norm of $f(A)=5A$ in 1 dimension. It fails with Stuck at the edge of dual feasibility. Any suggestions?

Constraints
$$
A succ 0\
Isucc A \
x I succ -5 A
$$

Objective

$$
text{min}_{A,x} x
$$

d = 1;
ii = IdentityMatrix(d);
(* Symbolic symmetric d-by-d matrix *)
ClearAll(a);
X = 5*ii;
A = Array(a(Min(#1, #2), Max(#1, #2)) &, {d, d});
vars = DeleteDuplicates(Flatten(A));

cons0 = VectorGreaterEqual({A, 0}, {"SemidefiniteCone", d});
cons1 = VectorGreaterEqual({ii, A}, {"SemidefiniteCone", d});
cons2 = VectorGreaterEqual({x ii, -X.A}, {"SemidefiniteCone", d});
SemidefiniteOptimization(x, cons0 && cons1 && cons2, {x}~Join~vars)
```

dg.differential geometry – Vector fields whose flows have constant norms

$newcommand{tr}{operatorname{tr}}$

Let $(M,g)$ be a smooth two-dimensional Riemannian manifold with boundary, and let $X$ be a vector field on $M$. Let $psi_t:M to M$ be the flow of $X$.

Suppose that $|(dpsi_t)_p|^2=langle (dpsi_t)_p,(dpsi_t)_p rangle=tr_g ((dpsi_t)_p^T(dpsi_t)_p)$ is independent of $p$, but not constant in $t$. Must $X$ be a homothetic vector field? i.e. does $L_x g=lambda g$ for some constant $lambda$? (In that case $psi_t^*g=e^{lambda t}g$. $lambda=0$ corresponds to Killing fields).

Since $f(t,p):=|(dpsi_t)_p|^2=tr_gbig((psi_t^*g)_pbig)$
a necessary and sufficient condition on $X$ is that $frac{partial }{partial t}f(t,p)=tr_g(psi_t^*L_Xg)$ would be independent of $p$, for all $t$.

In particular, for $t=0$, $tr_g(L_Xg)=2text{div}(X)$ should be constant.

So, a necessary (but insufficient) condition is for the divergence of $X$ to be constant.

linear algebra – Recover approximate monotonicity of induced norms

Let $A$ some square matrix with real entries.
Take any norm $|cdot|$ consistent with a vector norm.

Gelfand’s formula tells us that $rho(A) = lim_{n rightarrow infty} |A^n|^{1/n}$.

Moreover, from (1), for a sequence of $(n_i)_{i in mathbb{N}}$ such that $n_i$ is divisible by $n_{i-1}$, we also know that the sequence $|A^{n_i}|^{1/n_i}$ is monotone decreasing and converges towards $rho(A)$. I am interested in what happens when this divisibility property is not verified.

  1. If the matrix has non-negative entries, it seems the general property holds: For integers $n$ and $m$ such that $m > n$, it is the case that $|A^m|^{1/m} leq |A^n|^{1/n}$.

  2. If the matrix can have positive and negative entries, this more general observation does not seem to hold. I am trying to understand why it fails, how worse can the inequality become, and if it is possible to recover an inequality up to some function of $A$: $|A^m|^{1/m} leq f(A)cdot|A^n|^{1/n}$.

Any references to 1., or pointers for understanding 2. would be much appreciated.

(1) Yamamoto, Tetsuro. “On the extreme values of the roots of matrices.” Journal of the Mathematical Society of Japan 19.2 (1967): 173-178.

functional analysis – Relation between fractional and integer Sobolev norms

I encountered a situation where I have to add two norms defined on the boundary:

$$C_1 ||u||^2_{L_2(partialOmega)} + C_2 ||u||^2_{H^{3/2}(partial Omega)},$$

but do not really know how to manipulate this expression. Is there a relation between the norms?

Detection of integers that are norms of algebraic integers.

Leave $ K / mathbb {Q} $ be a numeric field and $ mathfrak {a} $ a comprehensive ideal of $ K $ of the rule $ n $. I'm trying to better understand how the ideal class group and narrow ideal class group can be used to detect integers in $ mathbb {Z} $ which can be written as norms of algebraic integers of $ K $.

Question 1 Am I correctly understanding that if $ mathfrak {a} $ is in the main class of the ideal class group either $ n $ or $ -n $ can be represented as the norm of an algebraic integer of $ K $ ?

Question 2 Now suppose that $ mathfrak {a} $ It is located in the main class of the narrow ideal class group. Am I correct in my understanding that then $ n $ is the norm of an algebraic integer of $ K $ ? More important to me: is this a yes and only if? I'm a little confused about the impact of the requirement that each positive inclusion be positive.

python: trying to trace the unitary sphere of R ^ 2 with different norms

I use this code:

import numpy as np
import numpy.linalg as nla
import matplotlib.pyplot as plt


#Definition of p-norm
def norma_p(vect, p):
    vect_abs = (abs(vect(i)) for i in range(len(vect)))
    vect_to_p = (vect_abs(i)**p for i in range(len(vect_abs)))
    return sum(vect_to_p)**(1/p)

#Obtaining 100 equidistant vectors with coordinates between (-1,1)
vecs = np.linspace((-1, -1), (1, 1), 100)

#Calculating each vector's norm in different norms
mags_1 = (nla.norm(vecs(i), 1) for i in range(len(vecs)))
mags_inf = (nla.norm(vecs(i), np.infty) for i in range(len(vecs)))
mags_euc = (nla.norm(vecs(i), 2) for i in range(len(vecs)))
mags_p1_5 = (norma_p(vecs(i), 1.5) for i in range(len(vecs)))
mags_p2 = (norma_p(vecs(i), 2) for i in range(len(vecs)))
mags_p3 = (norma_p(vecs(i), 3) for i in range(len(vecs)))

#attempt to make unitary vectors
uvecs_1 = (vecs(j)/mags_1(j) for j in range(len(vecs)))
uvecs_inf = (vecs(j)/mags_inf(j) for j in range(len(vecs)))
uvecs_euc = (vecs(j)/mags_euc(j) for j in range(len(vecs)))
uvecs_p1_5 = (vecs(j)/mags_p1_5(j) for j in range(len(vecs)))
uvecs_p2 = (vecs(j)/mags_p2(j) for j in range(len(vecs)))
uvecs_p3 = (vecs(j)/mags_p3(j) for j in range(len(vecs)))

to try to see how the unitary sphere changes with different norms, but instead of getting something like this

I keep getting the same plot for each of the supposedly "unitary" vectors
enter the description of the image here

Number number theory: algebraic numbers with small norms

For any complex number other than zero $ z_1, dotsc, z_m $, there are infinities $ n $is such that the arguments of $ z_1 ^ n, dotsc, z_m ^ n $ everyone lies in $ (- pi / 4, pi / 4) $. This follows from Dirichlet's theorem on the simultaneous diofanthin approach. For such $ n $s
$$ | z_1 ^ n + dotsb + z_m ^ n | geq Re (z_1 ^ n + dotsb + z_m ^ n) geq frac {| z_1 | ^ n + dotsb + | z_m | ^ n} { sqrt {2}}. $$
In particular, the left side cannot be asymptotically $ 1 / n! $, because the right side is exponentially small in the worst case.

In short, there is no $ alpha $ Meet the requirements.