0
votes
0answers
16 views

Comparing notions of continuity

I have trouble distinguishing 3 different types of continuity Uniform continuous Sequential continuous Equicontinuous Could someone explain the difference between 3 and give some examples? I am ...
1
vote
0answers
31 views

Extending linear continuous functions.

Let $E$, $F$ be normed vector spaces and $M$ a subspace of $E$. I'm trying to find an example of a function $f:M\to F$ such that $f$ is linear and continuous but that you can't extend it to ...
0
votes
0answers
12 views

From essential oscillation to a continuous representative

Let $u$ be a measurable function such that for every(former: a.e.) $x\in \Omega$ there holds for sequences $R_n,\delta_n\to 0$ that$$\omega_n:=ess-osc_{B_{R_n}(x)} u\leq \delta_n. \tag{1}$$ Edit: ...
2
votes
0answers
24 views

continuity of bilinear

Let $B: E\times F\rightarrow G$ be a continuous bilinear map of normed spaces, where $\|(e,f)\| = \|e\| _E+\|f\|_F$. Show that $\dfrac{\|B(e,f)\|}{\|(e,f)\|} \rightarrow 0$ as $(e,f) \rightarrow 0$. ...
1
vote
0answers
24 views

On characterization of Riesz homomorphisms on $C(X)$ space

How to prove the following: Let $K$ be an arbitrary topological space and $\pi: C(K)\to\mathbb R$ be a map with $\pi (1) = 1$. If $\pi$ is a algebra homomorphism then it is an Riesz homomorphism.
0
votes
1answer
19 views

Definitions of hemicontinuity

can anyone see the equivalence or relation between the following two definitions of hemicontinuity that I encountered: Assume that $K$ is a closed, convex subset of Banach space $X$. Let $X^{*}$ be ...
1
vote
1answer
59 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. ...
2
votes
1answer
34 views

What is the definition of this set of absolutely continuous function

I know that $$AC(a,b):=\left\{f \in C(a,b)|f(x) = f(c)+\int_c^x g(t) d \lambda(t),c \in (a,b), g \in L^1_{\text{loc}}(a,b)\right\}$$ $$AC[a,b]:=\left\{f \in C[a,b]|f(x) = f(c)+\int_a^x g(t) d ...
1
vote
1answer
25 views

Solutions depending on something continuously

Let $V$ be a a real Banach space, $K \subset V$ a closed convex set, $A: K \rightarrow V^{*}$ a (nonlinear) operator and $F \in V^{*}$. Then the variational inequality is the following problem: find ...
0
votes
2answers
54 views

Is my proof correct? Finite-dimensional normed vector spaces

I'm trying to prove that every finite-dimensional normed space is topological isomorphic to $\mathbb{R}^n$. Let $(E,\|\cdot\|_E)$ such that $dimE=n$ and let $$ T:\mathbb{R}^n\to E\\ x\mapsto ...
0
votes
1answer
29 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 ...
0
votes
0answers
18 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
40 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
66 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 ...
2
votes
1answer
58 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
27 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
29 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
29 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
85 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
44 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
62 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
20 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
42 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
48 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
48 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
28 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
49 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
42 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
82 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
111 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
54 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
22 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
36 views

Kernel closed implies continuous operator?

Is closed kernel sufficient for linear operators to be continuous? Counterexample? Thx, Alex
0
votes
0answers
24 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
45 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
58 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
388 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
63 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
74 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
137 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.