## Inequality on difference of norms squared

Let $$x, y$$ be two vectors. Is there a vicinity of $${x=y}$$ s. t. $$| x – y | ge |x|^2 – |y|^2$$?

## functional analysis – Equivalent norms in the intersection

Let $$V$$ be a vector space. Two norms $$|cdot |_1,|cdot |_2: V longrightarrow mathbb{R}$$ in $$V$$ are said equivalent if there exists $$a,b>0$$ such that
$$a|u|_1 leq |u|_2 leq b |u|_1,; forall ; u in V.$$

Now,, consider $$X=(X, |cdot|_X)$$ and $$Y=(Y, |cdot|_Y)$$ two normed spaces. It is easy to show that $$Z:= X cap Y$$ is a normed space with norm $$|cdot|_Z: Z longrightarrow mathbb{R}$$ given by
$$|u|_Z=|u|_X+|u|_Y,; forall ; u in Z.$$

Question. If, for some some $$alpha, beta>0$$, we consider $${rm N}: Z longrightarrow mathbb{R}$$ as
$${rm N}(u)=alpha |u|_X +beta |u|_Y,; forall ; u in Z$$
then $${rm N}$$ defines a equivalent norm (in $$Z$$) with respect the norm $$|cdot|_Z$$?

I don’t see, for instance, how to prove that
$$alpha |u|_X +beta |u|_Y leq gamma( |u|_X + |u|_Y),; forall ; u in Z tag{1}$$
for some $$gamma>0$$. Can I to use some property of $$max$$ or $$min$$? Or, can I only prove $$(2)$$ instead of the equivalence of norms? The equivalence is more stronger than $$(2)$$.

## Was a quotient of two norms considered as a constraint to a convex optimization problem before?

I want to solve the optimization problem
$$text{minimize }g(x) quad text{subject to} quad Vert xVert_{infty}/Vert xVert_{2} le s$$
for $$xinmathbb{R}^d$$ and $$sin(0,infty)$$.
The function $$g$$ is (strongly) convex and Lipschitz smooth.

I know, that I could probably try to find saddle points of the corresponding Lagrangian but I would like to know, if there is a faster or more elegant way.

Do you know of a similar problem, that has been considered before?

## norms – What are the functions such that \$ lVert f + g rVert_p^p = lVert f rVert_p^p + lVert g rVert_p^p\$?

Let $$1 leq p leq 2$$. I am looking for a characterization of the couples $$(f,g)$$ of functions $$f,g in L_p(mathbb{R})$$ such that
$$lVert f + g rVert_p^p = lVert f rVert_p^p + lVert g rVert_p^p.$$

For $$p = 2$$, this relation is satisfied if and only if $$langle f, g rangle = 0$$. For $$p = 1$$, it has been shown in this post that the condition is equivalent to $$f g geq 0$$ almost everywhere.
For a general $$p$$, the relation is clearly satisfied as soon as the product $$fg=0$$ almost everywhere. Is this latter sufficient condition also necessary?

If not, then what if we reinforce the condition with
$$lVert alpha f + beta g rVert_p^p = lVert alpha f rVert_p^p + lVert beta g rVert_p^p$$
for any $$alpha, beta in mathbb{R}$$?

## fa.functional analysis – Estimating certain tensor norms on Banach spaces

Let $$X$$ and $$Y$$ be Banach spaces. An operator $$u:Xto Y$$ s called nuclear if $$u$$ can be written as $$u=sum_{n=1}^infty x_n^*otimes y_n$$ with $$(x_n^*)subseteq X^*$$, $$(y_n)subseteq Y$$ such that $$sum_{n=1}^infty|x_n^*||y_n| Define $$N(u):=inf{sum_{n=1}^infty|x_n^*||y_n|}$$ infimum being taken over all representations. Denote $$C(n):=sup{N(BA):|A|_{ell_1^ntoell_infty^n}leq 1, |B|_{ell_infty^ntoell_infty^n}leq 1}.$$ Is $$suplimits_{ngeq 1}C(n)?

## gn.general topology – Do topologically equivalent norms have the same \$C^n(V)\$?

If $$lVertcdotrVert_1$$ and $$lVertcdotrVert_2$$ are topologically equivalent norms on a vector space $$V$$, and $$f:V rightarrow R$$, is it true that $$f$$ is $$C^n$$ when differentiated with $$lVertcdotrVert_1$$ implies $$f$$ is $$C^n$$ when differentiated with $$lVertcdotrVert_2$$?

I’m not sure where to begin, but one gets a sense if $$C^infty$$-ness is defined for manifolds which only have open sets, then the derivative of $$f$$ might only depend on the open sets and not choice of norm.

## banach spaces – Uniform smoothness and twice-differentiability of norms

To get to the simplest case, consider a norm $$|cdot|$$ over $$R^n$$ that is uniformly convex of power-type 2, that is, there is a constant $$C$$ such that $$frac{|x+y| + |x – y|}{2} le 1 + C |y|^2$$ for all $$x$$ with $$|x| = 1$$ and for all $$y$$.

Question: Does this guarantee that $$|cdot|$$ has a second-order Taylor expansion on $$R^n setminus {0}$$, that is, there is a vector $$g$$ and a symmetric matrix $$A$$ such that $$|x + y| = |x| + langle g, y rangle + frac{1}{2} langle Ay, y rangle + o(|y|^2)$$ for all $$x neq 0$$. (Apparently this is a weaker requirement than twice-differentiability of $$|cdot|$$ on $$R^n setminus {0}$$)

It is easy to see that $$|cdot|$$ is differentiable on $$R^n setminus {0}$$, and a classic result of Alexandrov guarantees that the above second-order Taylor expansion holds for any convex function on almost every point $$x$$. It is also known that the norm of any separable Banach space can be approximated arbitrarily well by a power-type 2 norm that is twice differentiable on $$R^n setminus {0}$$ (see Lemma 2.6 here). But I wonder if the original norm itself has a second-order Taylor expansion.

## linear algebra – Frobenius and operator norms of a certain structured matrix

$$defp#1#2{frac{partial #1}{partial #2}}defR{mathbb R}$$

Let $$X in mathbb R^{n times d}$$ and $$y in mathbb R^{n times 1}$$ and $$u$$ be a unit vector in $$mathbb R^{k times 1}$$. Define the function $$F:mathbb R^{d times k} to mathbb R$$ by $$F(W) := y^top psi(XW)u$$ for some twice-differentiable function $$psi:mathbb R to mathbb R$$, and $$psi(A)$$ is the matrix with $$ij$$th entry $$psi(a_{i,j})$$ (i.e element-wise application of \$psi).

For any $$W in mathbb R^{d times k}$$, consider the the hessian $$nabla^2 F(W) in mathbb R^{dk times dk}$$.

Suppose $$|psi”|_infty := sup_{t in mathbb R} |phi”(t)| le alpha$$, $$|X|_{op} le beta$$, and $$|y| le gamma$$.

Question What are good upper-bounds for $$sup_W |nabla F(W)|_F$$ and $$sup_W |nabla^2 F(W)|_{op}$$ in terms of $$alpha$$, $$beta$$, and $$gamma$$ ?

Define
Define the matrices
eqalign{ Z &= XW qquad&impliesquad dZ = X,dW \ P &= psi(Z) qquad&impliesquad dP = Qodot dZ \ Q &= psi'(Z) qquad&impliesquad dQ = Rodot dZ \ R &= psi”(Z) \ }
where $$(odot)$$ denotes the elementwise/Hadamard product and $$,(psi’,psi”),$$ denote the ordinary first and second derivatives of the $$psi$$ function.

In this post, it has been shown that

If $$H := nabla^2 F(W)$$ and $$:$$ denotes trace inner product, then

eqalign{ {cal H}_{j,j,ell,ell’} &= x_ell^Tbig(r_jodot yodot x_{ell’}big);u_jdelta_{j,j’} \ }
where the vectors $$(x_ell,r_ell)$$ are the $$ell^{th}$$ columns of the $$(X,R)$$ matrices.
\$\$

## A bound for the Frobenius / Hilbert-schmidt norm

$$begin{split} |H|_F^2 &= sum_{l=1}^dsum_{l’=1}^dsum_{j=1}^k((y odot x_l)^top(r_j odot x_{l’}))^2u_j^2 le sum_{l=1}^dsum_{l’=1}^dsum_{j=1}^k u_j^2|y odot x_l|^2|r_j odot x_{l’}|^2\ &le sum_{l=1}^dsum_{l’=1}^d|y odot x_l|^2sum_{j=1}^k u_j^2|r_j odot x_{l’}|^2 le sum_{l=1}^dsum_{l’=1}^d|y odot x_l|^2sum_{j=1}^k u_j^2|r_j odot x_{l’}|^2 \ &le sum_{l=1}^d|y odot x_l|^2sup_{j=1}^k sum_{l’=1}^d|r_j odot x_{l’}|^2 = |X^top y|^2sup_{j=1}^k|X^top r_j|^2\ & le |X|_{op}^4|y|^2sup_{j=1}^k|r_j|^2 = beta^4gamma^2sup_{j=1}^k|r_j|^2 le alpha^2beta^4gamma^2. end{split}$$

Therefore $$|nabla^2 F(W) |_F le alphabeta^2gamma$$ for all $$W in mathbb R^{d times k}$$.

## Expressing the difference of uniform norms of two vectors in terms of difference of vectors

For two vectors $$x$$ and $$y$$ in $$d$$-dimensionsal Euclidean space, is there a way to write $$|x|_infty-|y|_infty$$ as a function of some sort of the vector $$(x-y)$$? I haven’t been able to find any resources which give any insight into finding such an expression.

## oc.optimization and control – Are all corecive norms obtained from bilinear forms on Hilbert spaces

Let $$H$$ be a Hilbert space with inner-product induced norm $$|cdot|_H$$. If $$|cdot|$$ is some other coercive norm on $$H$$, i.e.:
$$limlimits_{|x|_Htoinfty} |x|=infty.$$
then does there necessary exist a lower-semi-continuous symmetric Bilinear form $$B:Htimes Hrightarrow (0,infty)$$ satisfying
$$|x|=sqrt{B(x,x)}?$$