# Tagged Questions

50 views

### $C^0$ is a closed subspace of $L^{\infty}$

Let $\Omega\subset\mathbb{R}^n$ be an open bounded set. Let $f\in C^0(\bar\Omega)$. I have to prove that $\|f\|_{\infty}=\|f\|_{L^{\infty}}$. One implication is trivial. Let's consider the other one. ...
34 views

27 views

### Continuity of a map to a Frechet space

Let $(A,\| \cdot \|)$ be a normed space and $B$ be a Frechet space equipped with a family $\{ p_k \}_{k \in \mathbb{N}}$ of seminorms. Let $\phi: A \to B$ be a linear transformation satisfying the ...
16 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 $x\mapsto f(x,\cdot)$ ...
39 views

13 views

25 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 ...
26 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 ...
25 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 ...
81 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 ...
42 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 ...
59 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 ...
18 views

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 ...
39 views

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?
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 $T$ ...
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. ...
41 views

80 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 ...
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.
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 ...
104 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 $W^{1,\infty}(I)$ ...
39 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 ...
21 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, ...
30 views

### Kernel closed implies continuous operator?

Is closed kernel sufficient for linear operators to be continuous? Counterexample? Thx, Alex
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 ...
44 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 ...
52 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 $L^{\infty}$ ...
301 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 ...
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 ...
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 ...
73 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?
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 $M_1$ $$d_1(x,y)<\delta \implies ... 1answer 41 views ### Prove that this operator is continuous [duplicate] Let X,Y,Z be Banach spaces, and let T:X\to Y be linear. Let J:Y\to Z be linear, bounded and injective. If JT:X\to Z is bounded, then T is bounded. 1answer 70 views ### Convergence in Sobolev Spaces Consider the bounded mapping A:W^{1,p}(\Omega) \rightarrow W^{1,p}(\Omega)^{*} where A is defined as: \langle A(u),v \rangle\text{ } := \int_{\Omega}a(x,u,\nabla u)\cdot \nabla v + c(x,u,\nabla ... 1answer 116 views ### Weak continuity in Sobolev Spaces First consider the following two Sobolev Embedding Theorems. Theorem 1: The continuous embedding W^{1,p}(\Omega) \subset L^{p^{*}}(\Omega) holds provided the exponent p^{*} is defined as ... 1answer 128 views ### x_n convergence to x implies f_n(x_n) convergence to f(x). prove that f is continuous Let f and f_n be functions from \mathbb{R} \rightarrow \mathbb{R} Assume that f_n (x_n) \rightarrow f (x) as n\rightarrow \infty whenever x_n \rightarrow x. Prove that f is ... 1answer 64 views ### Continuity of double centralizers in Banach algebras I had some problems with a certain exercise, came up with a solution, but I'm not sure it is correct. Exercise ("MURPHY, C*-Algebras and Operator Theory", Chapter 2, exercise 1) Let A be a ... 1answer 53 views ### Different types of continuity in \ell^2 Consider the following functional J on \ell^2 which for x = \{x_n\} is defined by$$J(x) = \sum_{n=1}^{\infty}n^{1/n}x_{n}^{2}. Is $J$ continuous? Is $J$ lower semi-continuous? Is $J$ ...
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 ...