0
votes
0answers
15 views

Relation between continuity as a map and joint continuity

Let $f=f(x,y) : \mathbb{R}^2 \to \mathbb{R}$ and denote by $C(\mathbb{R})$ the space of bounded and continuous, real-valued functions on $\mathbb{R}$. Is it true that if the map $x\mapsto f(x,\cdot)$ ...
1
vote
1answer
38 views

The continuous dual of the reals

I just have a few questions involving the continuous dual of $\mathbb{R}^{N}$. We know that the dual $(\mathbb{R}^{N})^{*}$ of $\mathbb{R}^{N}$ is the space of all linear forms $$a: \mathbb{R}^{N} ...
0
votes
0answers
45 views

implicitly define a function

The first part i made $u=\frac{z}{x}$ and $v=\frac{y}{x}$ and after calculating the partial derivatives $\frac{dz}{dx}$ and $\frac{dz}{dy}$ The second i have no idea how to do it
0
votes
1answer
67 views

The linearity of $D \beta : \mathbb{E_1} \times \mathbb{E_2} \rightarrow \mathcal{L}(\mathbb{E_1} \times \mathbb{E_2},F)$

Let $\mathbb{E_1}, \mathbb{E_2}$ and $\mathbb{F}$ normed spaces of finite dimensions and $\beta : \mathbb{E_1} \times \mathbb{E_2} \rightarrow \mathbb{F}$ is one bilinear function. Then $D \beta : ...
2
votes
2answers
63 views

Is $T':L^2(\Omega) \to L^2(\Omega)$ continuous?

Here, $k$ is a fixed number. Let $$T(x) = \begin{cases} -k &x \in (-\infty, -k]\\ x &x \in (-k, k)\\ k &x \in [k, \infty) \end{cases}.$$ So $$T'(x) = \begin{cases} 0 &x \in (-\infty, ...
1
vote
0answers
13 views

Finding the continuity of the mapping of a solution to a PDE to its partial derivative

Here is a modified version of the Black-Scholes PDE: $\frac{\partial \phi(t,S,i)}{\partial t}$ + $r_iS\frac{\partial \phi(t,S,i)}{\partial S}$ + $\frac{1}{2} \sigma^2_i S^2 \frac{\partial^2 ...
1
vote
0answers
46 views

Is continuous extension on dense subset an isometry

If we have that $X \subset V$ is dense linear subspace. Where $V$ is normed space. I can show that for any $f \in X^{*}$, there exists a unique extension $\bar{f}$. I want to know if it can be shown ...
1
vote
2answers
20 views

Investigating a function with a parameter

I got stuck on solving this problem: For which $a \in \Bbb R$ is the function $$ f_a: \ ]1, \ \infty[ \; \longrightarrow \ \Bbb R: x\mapsto \frac{\log x}{(x-1)^a} $$ continuous on $[1, \ ...
2
votes
0answers
24 views

extension theorems on normed spaces

I know that there are a number of extension theorems, Tietze's extension theorem, Hahn-Banach extension and so on.. I want to know if there is an extension theorem which guarantees that if say $X$ is ...
1
vote
1answer
24 views

Is the Inverse of the Vectorised Solid Angle Equation for $n$ Circular Discs Continuous?

I have a continuous function$^{*1}$ that takes in 3 arguments, and returns 24 outputs. I want to know if the inverse of this function is continuous. The 3 input arguments are the x, y, and z position ...
0
votes
1answer
18 views

Does weakly differentiable and $L^{\infty}$ imply continuity

Suppose $\Omega \subset \mathbb{R}^d$ is open, connected and bounded. Is $$W^{1,1}(\Omega)\cap L^{\infty}(\Omega) \subset C(\bar{\Omega})?$$ Here $W^{1,1}$ denotes the space of all weakly ...
4
votes
1answer
80 views

Continuous function on closed unit ball

Take a continuous mapping $f: \bar{B^{n}} \rightarrow \bar{B^{n}}$, where $\bar{B^{n}}$ is a closed unit ball in $\mathbb{R}^{n}$. Assume that $f(x) \neq x$ for every $x \in \bar{B^{n}}$. Define ...
1
vote
0answers
39 views

Measure theory integration question involving continuous function

Quick measure theory question. Given that $\Omega \subset \mathbb{R}^{n}$ and $f$ is continuous on $\Omega$. How would you show that if $$\int_{\Omega}f \, dx = 0$$ Then $f = 0$ everywhere? Thanks ...
4
votes
2answers
58 views

A linear operator between $C[0,1]$ and $C[0,1]$ defined as $Tf = f + \int f$; Show $T$ is an isomorphism

Define a linear operator $T:C[0,1] \to C[0,1]$ as follows: $$Tf(x) = f(x) + \int_0^x f(u)du$$ It is easy to show that $T$ is a bounded linear operator. The statement also (1) claims that $T$ is ...
0
votes
1answer
18 views

Please verify my work about an equicontinuous sequence

Please check this work below. It is self-explanatory. I am unsure because I use a sequence composed with another sequence with the same index ($f_n^{-1}(u_n)$). We have a sequence of functions ...
0
votes
1answer
39 views

Are these $f_n$ equicontinuous?

Let $f_n$ be a sequence of real-valued functions defined on $\mathbb{R}$ satisfying $f_n \to f$ uniformly in the compact subsets of $\mathbb{R}$ $f_n^{-1}$ is bi-Lipschitz $1 \leq (f_n^{-1})'(x) ...
1
vote
0answers
28 views

Biorthogonal functionals continuous? [duplicate]

If I have a Schauder basis $(x_n)$ of a Banach space $X$. Such that for every $x = \sum_{i=1}^{\infty} a_i x_i$ for a unique sequence $(a_i) \subset \mathbb{R}$. Is it obvious that the functionals ...
3
votes
1answer
39 views

Proof of compactness of Lipschitz functions

Consider the set $\mathcal{F}$ of continuous functions on $[0;1]$ with boundary values $$ f(0)=f(1)=0 \qquad \forall f \in \mathcal{F}. $$ Define the metric $d(f,g) = \lVert f-g \rVert_\infty = ...
0
votes
1answer
39 views

The difference between semicontinuity and hemicontinuity.

For a point-to-set function F, is "upper hemicontinuous" the same as "upper semicontinuous"? If not, then what's the difference?
0
votes
2answers
27 views

Continuity of an operator in $C^0[0,1]$ with different norm

Let $C^0[0,1]$ be the space of real valued continuos functions with the norm $\|f\| = \int \limits_{0}^1 x^2 |f(x)| dx$ and let $T \colon C^0[0,1] \to C^0[0,1]$ such that $f(x) \mapsto f(1-x)$. Is $T$ ...
3
votes
2answers
48 views

Examples of Functions

Alright so I am trying to find examples of functions that are differentiable at a point, but not continuous there. Also a function continuous at no point; a function continuous only at one point. ...
2
votes
1answer
41 views

Examples of contractions between functional spaces

Define $\mathcal{F}$ as the following set of continuous functions: $$ \mathcal{F} := \left\{ f: \mathbb{R} \rightarrow \mathbb{R}^n \mid f(\cdot) \ \text{contin.}, \ f(x) \in K(x) \subset ...
1
vote
1answer
28 views

Continuity of an application between function spaces.

I'm trying to prove the following statement... Let $f:[a,b] \times \mathbb{R} \to \mathbb{R}$ a bounded and continuous function, $t_{0} \in [a,b]$, $x_{0} \in \mathbb{R}$, $r>0$ and $$B= \{ x ...
3
votes
0answers
78 views

A problem of weak* continuity in relation with semigroups

Let $(\Omega,\Sigma,\mu)$ be a probability space. Let $\mathcal{A}$ ba a $\sigma$-subalgebra of $\Sigma$. We denote by $\mathbb{E} \colon L^\infty(\Sigma) \to L^\infty(\mathcal{A})$ the associated ...
1
vote
2answers
40 views

Show that a map is a continuous bilinear form on $H^1(0,1)$ space

Let $u,v \in H^1(0,1) = \{f : (0,1) \longrightarrow \mathbb{R}, f,f' \in L^2(0,1) \}$, show that $$a(u,v) = \int_0^1 (u'v' + uv)\; dx$$ is a continuous bilinear form.
0
votes
0answers
42 views

Seemingly easy analysis problem but unsure how to proceed.

if $f(x)=\frac{1}{x+2}$ then $f(x)=1-(x+1)+(x+1)^2+T$ for some $x_0$ between $x$ and $-1$ where $T=-\frac{(x+1)^3}{(2+x_0)^4}$ I'm not sure how to proceed in solving this problem. We recently ...
4
votes
1answer
98 views

Showing Lipschitz continuity of Sobolev function

Is there any problem with the following, please advise: Take $I \subset \mathbb{R}^{n}$ convex, closed and bounded. I want to show that if I have $u_{m} \rightharpoonup^{*} u$ in $W^{1,\infty}(I)$ ...
1
vote
0answers
34 views

about a theorem of weakly lower semicontinuous functions

I am studying the proof of the following theorem Theorem: Let $E$ a Hilbert space and suppose that $\varphi :E \rightarrow R$ is a weakly lower semicontinuous functional. Suppose that $\varphi$ is ...
1
vote
0answers
20 views

Question in the Continuity of a function

I have this function: $$(J''(u)v,w)=(v,w)-(KN_{f'}(Ku)Kv,w)$$ for all $u,v,x\in L^2[0,1]$ such that $f\in C^1([0,1]\times\mathbb{R},\mathbb{R})$ and $Ku(t)=\int_0^1 G(t,s) u(s)ds$ $K$ is symetric, ...
0
votes
1answer
29 views

Kernel closed implies continuous operator?

Is closed kernel sufficient for linear operators to be continuous? Counterexample? Thx, Alex
0
votes
0answers
22 views

Regularising a function that is constant on an interval (related to Heaviside)

Define the function $f:\mathbb R \to \mathbb R$ by $$f(x) = \begin{cases} x &\text{for $x < 0$}\\ 0 &\text{for $x \in [0,1]$}\\ x-1 &\text{for $x > 1$}& \end{cases} $$ Note that ...
2
votes
1answer
43 views

Weak Lower Semicontinuity Generalized to any $L_{p}$ space

I am having difficulty with the following proof: Generalize the weak lower semi-continuity of$L^{p}$ norms to all $1\leq p < \infty$; i.e., show that if $u_{n}\to u$ weakly in $L^{p}$, then ...
2
votes
2answers
51 views

$C^0(\overline{\Omega}) \subset L^{\infty}(\Omega) ?$

In my functional analysis script there is an example that uses $$C^0(\overline{\Omega}) \subset L^{\infty}(\Omega) $$ where $\Omega \subset \mathbb{R}^n$ is an open subset and we take $L^{\infty}$ ...
6
votes
1answer
282 views

Equicontinuity implies uniform convergence

So I know it's a theorem that if $\{f_n\}$ is a sequence in an equicontinuous family of functions defined on a compact metric space $K$ then if for all $x$, $f_n(x)\rightarrow f(x)$ pointwise then ...
2
votes
0answers
62 views

Space of Continuous mappings to metric spaces

I want to ask whether some basic result from the space $C([0,1],R)$, where $R$ is the real space carries over to the space $C([0,1],E)$, where $(E,\|\cdot\|_E)$ is a metric space. We know that ...
2
votes
2answers
47 views

how to prove that the limit of this sequence of functions is continuous?

I have a norm that works in function space and that is $‖∙‖_{sup}:C([0,1])→R$, $‖∙‖_{sup}:=sup${$|f(t)|$}. I need to show that the metric is complete. So I need to show that every Cauchy sequence of ...
1
vote
1answer
72 views

addition and multiplication of functions in function space, continuous?

I have a norm that works in function space of C[0,1]. How do I show that addition and multiplication of functions (C[0,1]xC[0,1]->C[0,1]) are continuous functions?
1
vote
3answers
130 views

Can $\le$ be used insted of < in the definition of continuity?

A common definition of a continuous map $T:M_1\to M_2$ is that for every $x\in M_1$ and every $\epsilon>0$ there exists a $\delta >0$ such that for all $y$ in $M_1$ $$d_1(x,y)<\delta \implies ...
0
votes
1answer
41 views

Prove that this operator is continuous [duplicate]

Let $X,Y,Z$ be Banach spaces, and let $T:X\to Y$ be linear. Let $J:Y\to Z$ be linear, bounded and injective. If $JT:X\to Z$ is bounded, then T is bounded.
2
votes
1answer
70 views

Convergence in Sobolev Spaces

Consider the bounded mapping $A:W^{1,p}(\Omega) \rightarrow W^{1,p}(\Omega)^{*}$ where $A$ is defined as: $\langle A(u),v \rangle\text{ } := \int_{\Omega}a(x,u,\nabla u)\cdot \nabla v + c(x,u,\nabla ...
3
votes
1answer
113 views

Weak continuity in Sobolev Spaces

First consider the following two Sobolev Embedding Theorems. Theorem 1: The continuous embedding $W^{1,p}(\Omega) \subset L^{p^{*}}(\Omega)$ holds provided the exponent $p^{*}$ is defined as ...
2
votes
1answer
128 views

$x_n$ convergence to $x$ implies $f_n(x_n)$ convergence to $f(x)$. prove that $f$ is continuous

Let $f$ and $f_n$ be functions from $\mathbb{R} \rightarrow \mathbb{R}$ Assume that $f_n (x_n) \rightarrow f (x)$ as $n\rightarrow \infty$ whenever $x_n \rightarrow x$. Prove that $f$ is ...
4
votes
1answer
63 views

Continuity of double centralizers in Banach algebras

I had some problems with a certain exercise, came up with a solution, but I'm not sure it is correct. Exercise ("MURPHY, C*-Algebras and Operator Theory", Chapter 2, exercise 1) Let $A$ be a ...
0
votes
1answer
53 views

Different types of continuity in $\ell^2$

Consider the following functional $J$ on $\ell^2$ which for $x = \{x_n\}$ is defined by $$J(x) = \sum_{n=1}^{\infty}n^{1/n}x_{n}^{2}.$$ Is $J$ continuous? Is $J$ lower semi-continuous? Is $J$ ...
3
votes
1answer
92 views

Using Nemytskii Theorem for Sobolev Spaces

The Nemytskii mappings in Lebesgue spaces theorem is as follows: If $a: \Omega \times \mathbb{R}^{m_{1}} \times \cdots \times\mathbb{R}^{m_{j}} \rightarrow \mathbb{R}^{m_{0}}$ is a Caratheodory ...
1
vote
2answers
42 views

Analogue of closed graph theorem

This is the analogue of closed graph theorem for compact space Suppose that $X$ and $K$ are metric spaces, that $K$ is compact, and that the graph of $f: X \rightarrow K$ is a closed subset ...
1
vote
0answers
21 views

Some basic questions about $C^k(I \times A)$

Let $I=[0,T]$ and $A$ be a bounded set that may or may not be compact. Let $f=f(x,t) \in C^k(I \times A)$. Am I right: 1) $f_x, f_t \in C^{k-1}(I \times A)$ 2) $f_{xt}, f_{tx}, f_{xx}, f_{tt} \in ...
0
votes
1answer
34 views

Showing a function is continuous on a compact set

Let $\Omega$ be bounded and open in $\mathbb{R}^n$. Let $f \in C^1([0,T]\times \Omega).$ If $f(t, \cdot) \in C^1(\overline{\Omega})$ for each $t$, how do I show that $f \in C^1([0,T]\times ...
2
votes
2answers
45 views

Compact operator and limit

I was wondering about something related to compact operators. If we have a compact operator $T:X \mapsto Y$ and a bounded sequence $(x_n)n$, then we know that there is a convergent subsequence ...
4
votes
1answer
35 views

Question about the continuity property of a function in topological vector space

I'm reading Functional Analysis book of Walter Rudin, and there's one point in this book that I don't know why he states that. Here is the statement: $f$ is a linear mapping from F-space $X$ into ...