Tagged Questions

2
votes
0answers
126 views

Space of functions that are everywhere differentiable

Define the space $\beta^1([a,b])$ as the space of functions $f : [a,b] \mapsto \Bbb R$ which are everywhere differentiable and whose derivative $f'$ is a bounded function. One equips this space with ...
2
votes
1answer
51 views

Question regarding Gâteaux Derivative

This is a question on an assignment for a grad engineering class that I cannot seem to figure out. The statement is as follows: Consider $X$ the space of continuous functions on the interval [0,1]. ...
4
votes
4answers
129 views

Proving $C_c^\infty(\Omega)$ is not complete

Let $\Omega \subseteq \mathbb{R}^n$ be a domain. How can I prove that $C_c^\infty(\Omega)$ (with the usual topology) is not sequentially complete? I don't think I have ever seen a proof of this ...
0
votes
1answer
25 views

How to prove this equality in proportional fairness analysis?

How to prove: $$\sum_{s=1}^S\left(\frac{y_s-x_s}{x_s}\right)=\bigtriangledown J_\vec x\cdot(\vec y-\vec x)$$ with: $$J_\vec x=\sum_s\ln(x_s)$$
1
vote
1answer
95 views

Normal convergence

I have some problems to apply normal convergence of series of functions in any vector space. In fact $(f_{n})$ is a sequence of differentiable functions defined from a topological space $X$ to a ...
1
vote
1answer
73 views

A question about the second differential

Hi I have a doubt: What is the matrix associated at the second differential? Let $f:\mathbb{R}^n \rightarrow \mathbb{R}$, differiantable and let $df: \mathbb{R}^n \rightarrow \mathbb{R}$, ...
1
vote
1answer
50 views

Identify the distrionbutional derivative with classical derivative?

I am reading Rudin's Functional Analysis and got quite confused by his proof for theorem 7.25, which he calls Sobolev's Lemma. In proving the theorem, he defines the function $F$, and calculates its ...
4
votes
2answers
230 views

Does the derivative of a continuous function goes to zero if the function converges to it?

Physicist here. I am puzzled by a question: looking at a continuous function $g :\mathbb{R} \rightarrow \mathbb{R}$ that goes to zero at infinity, I am interested in the behavior of its derivative ...
1
vote
2answers
364 views

Frechet derivative

I want to find the Fréchet derivative of the following functional: $$ \begin{align} F : C[-1,1] &\rightarrow \mathbb{R}\\ x &\mapsto x(0)\int_0^1 \sin\ x(t) \, dt. \end{align} $$ How can I do ...
0
votes
0answers
93 views

derivative of conditional expectation operators $\mathbb{E}_t$?

Let $(\mathcal{F}_t)_{t\in [0,T]}$ be a filtration on a probability space $\Omega$. Fix $1<p<\infty$. Let $\{\mathcal{E}(\cdot|\mathcal{F}_t)\ :\ t\in [0,T] \}$ the associated family of ...
2
votes
2answers
295 views

Functional Derivative

I am completely confused about calculating the (infinitesimal) change of an expression that depends on a function, when I vary the function. Consider the following: $$ ...
2
votes
0answers
115 views

rényi entropy as a derivative

Let $x=(x_i)$ be a probability measure on $\{1,\ldots,n\}$. Suppose $1<p<\infty$. The Rényi entropy of $x$ is $$ H^p(x)=\frac{1}{1-p}\log \sum_{i} x_i^p. $$ Does there exist a formula for ...
4
votes
1answer
1k views

Finding the derivative of the norm

Consider function from Hilbert space to real numbers. $F(x)=\| Ax\|$. My question how to find it's derivative $F'(x)$.
6
votes
2answers
1k views

Differentiating an Inner Product

If $(V, \langle \cdot, \cdot \rangle)$ is a finite-dimensional inner product space and $f,g : \mathbb{R} \longrightarrow V$ are differentiable functions, a straightforward calculation with components ...
0
votes
0answers
229 views

Weak derivative

Let $u \in C(\Omega)$ be a function with weak derivative $Du \in C(\Omega)^n$. How does one prove that $Du$ coincides with the classical derivative? Is the mean value theorem for integration ...