4
votes
3answers
65 views

Relation between continuous maps and convergence of sequences

I am studying metric spaces and I know that in a normed space $E$ a map $T:E \to E$ is contínuous if and only if $T(x_n) \to T(x)$ for every convergent sequence $x_n \to x$ in $E$. In my notes there ...
0
votes
2answers
52 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
28 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 ...
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} ...
2
votes
1answer
55 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 ...
0
votes
0answers
27 views

Show that $f: K_1(0) \rightarrow \mathbb{R}^3$ is Lipschitz.

Firstly, the Assignment: Let $V = (\mathbb{R}^3 ,\|\cdot\|_{\infty})$ where $\|\cdot\|_{\infty}$ denotes the maximum norm and consider the function: $$f: K_1(0) \rightarrow ...
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 ...
0
votes
0answers
38 views

Is this map continuous?

Let $C^1([0,1])$ denote the space of continuously differentiable functions on the interval $[0,1]$ with the supremum norm induced from $C([0,1])$. Is the following map continuous? $M_1:(C^1([0,1]), ...
3
votes
0answers
48 views

What extra assumption makes this transformation affine?

Let a vector space $V$ be given. Let $f:V\to V$ have the property that for all $x,y,a\in V$, $$ f(x+a)-f(y+a) = f(x) - f(y) \tag{$\star$} $$ Q1. I'd like to know how weak one can make additional ...
3
votes
1answer
189 views

Normed Vectors Spaces

Let $(E,\| \cdot \|_E)$ and $(F,\| \cdot \|_F)$ be two normed vector spaces over $\mathbb{C}$ and let $u: E\rightarrow F$ be a linear map. (a). Prove that the following conditions are equivalent: i. ...
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
1answer
72 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
134 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 ...
1
vote
1answer
143 views

Show C(X) is a vector space over $\mathbb R$ with the following operations?

I have a set of continuous functions, $C(X): X \rightarrow R$ on a compact metric space, and definitions of addition & multiplication: $$(f+g)(x) = f(x)+g(x)$$ $$(\lambda f)(x) = \lambda ...
0
votes
1answer
27 views

Determine all the $x_0$ such that $\phi : \mathbb C[X] \to \mathbb C, P \mapsto P(x_0)$ is continuous

In $\mathbb C[X]$, we consider the norm $\left\lVert P \right\rVert = \sup \left|a_i\right|$ for $P(X) = \sum_{i=1}^na_ix^i$. For all $x_0$ we consider the linear form $\phi : \mathbb C[X] \to \mathbb ...
0
votes
1answer
1k views

The definition of locally Lipschitz

I am given this definition: A function $f:A\subset\mathbb R^n\to\mathbb R^m$ is locally Lipschitz if for each $x_0\in A$, there exist constants $M>0$ and $\delta_0 >0$ such that ...
0
votes
1answer
29 views

Clarification about the space $C^0 ([-T,T],B)$

Let $B$ be a Banach space (in particular, $B$ is a function space equipped with the supremum norm). The space $C^0 ([-T,T],B)$ is the set of continuous functions on $[-T,T]$, valued in $B$. ...
0
votes
1answer
37 views

Is $D$ well-defined?

In my text there's a problem which reads as: Consider $C[0, 1]$ with the norm $\|.\|_\infty$. Let $Y$ be the vector subspace of all differentiable functions on $[0, 1].$ Consider the linear map ...
5
votes
1answer
469 views

Prove the boundedness of a bilinear continuous mapping.

Let $X,Y,Z$ are Banach spaces and $$B:X\times Y\to Z$$ is bilinear and continuous. Prove that there exists $M<\infty$ such that $$\lVert B(x,y)\rVert \leq M\lVert x\rVert\lVert y\rVert.$$ Is ...
1
vote
1answer
56 views

Unique continuos linear function given a continuous function from a dense space in X to Y (Y is a Banach Space).

Let $X$ be a normed space, let $Y$ be a Banach Space, let $D\subseteq X$ be a dense linear subspace of $X$ and let $L:D\rightarrow Y$ be a continuous linear function. Then there is a unique continuous ...
0
votes
2answers
67 views

Find a convergent function in metric space

Let $C[−1, 1]$ be the space of continuous functions equipped with the metric $p(f,g) = \max\{|f(x)−g(x)| \mid x \in [−1, 1]\}$. Then the sequence of functions $(f_n):[−1,1]\rightarrow \mathbb{R}$ ...
1
vote
3answers
69 views

Determining whether $f(x) = \frac{\sin||x||}{e^{||x||}-1}$ for $x \neq 0$, $f(x) = 1$ for $x = 0$ is continuous at $0$

$f: \mathbb R^m \to \mathbb R$ is defined as $$f(x) = \begin{cases}\dfrac{\sin||x||}{e^{||x||}-1} & \text{if $x \ne 0$} \\ 1 & \text{if $x = 0$.}\end{cases}$$ Note that $x$ is a vector in ...
2
votes
1answer
92 views

(p-q)-Lipschitz continuity of linear function

I have the following linear function $f(x,y,z) = ax + by + cz.$ I need to prove that f() is (p-q) Lipschitz continuous where $p=1$ and $q=\infty$. For a given two points $(x_1, y_1, z_1)$ and $(x_0, ...
1
vote
1answer
46 views

Continuity of $J: GL_c(E) \rightarrow GL_c(E): T \mapsto T^{-1}$

I have a question about a proof in my analysis textbook. They show that if $E$is a banach space, then $J: GL_c(E) \rightarrow GL_c(E): T \mapsto T^{-1}$ is continuous by first showing that it is ...
-4
votes
2answers
421 views

Continuity in a normed space

Let $X$ be a normed space. Show that the function $f:X \to R$ defined by $f(x)=\|x\|$ is continuous on $X$.
1
vote
1answer
384 views

Two weird proofs about continuity in normed vector spaces

I am reading a pair of "proofs" that a friend sent to me. I really don't understand some passages, so I hope someone could help me. The questions are the following First Question. The result to be ...