A vector space $E$, generally over the field $\mathbb R$ or $\mathbb C$ with a map $\lVert \cdot\rVert\colon E\to \mathbb R_+$ satisfying some conditions.
0
votes
1answer
26 views
Exercise of interior of a closed ball
Let $E$ a normed spaces and $a\in E$. How to prove that $$[\overline{B}(a, r)]^{\circ}\subseteq B(a, r)$$ where $\overline{B}(a, r)=\{x\in E: \|x-a\|\leq r\}$ and $B(a, r)=\{x\in E: \|x-a\|< r\}$
...
3
votes
1answer
45 views
Two terms that I want to understand: weakest topology and jointly continuous (in the following context).
I was reading an article online, please help me to understand the following lines (in bold letters). -
Topological structure:
If (V, ‖·‖) is a normed vector space, the norm ‖·‖ induces a metric and ...
0
votes
0answers
41 views
Does every Banach space admit a continous injection to a non-closed subspace of another Banach space?
Let $(V, ||\,||)$ be a Banach space. I want to produce a non-complete norm $||\,||'$ on it such that $||v||' \leq ||v||$ for all $v$ in $V$. Given a continuous injection $\varphi\colon V \to W$ with a ...
4
votes
2answers
83 views
Why doesn't the Weierstrass approximation theorem imply that every continuous function can be written as a power series?
I hope that my question in the title is well formulated.
I am a little bit confused with the next exercise from a book:
Argue that there exists functions $f \in C[0, \frac{1}{2}]$ that cannot be ...
0
votes
0answers
29 views
Why can I choose elements of $X$ (and $h \in L^2(0,T)$) in this way? (Dual spaces, norms, Bochner spaces)
This is from the book Vector Measures by Diestel and Uhl, page 98:
Let $X$ be a Hilbert space. Let $\epsilon > 0$ and suppose first that $g = \sum_{i=1}^\infty x_i^* \chi_{E_i}$ where $x_i^* ...
2
votes
3answers
78 views
Show that $c$ is closed in $l^{\infty}$
Let $$c=\{ (a_i)_{i \in \mathbb{N}} \ ; \ \ a_i \in \mathbb{R}\ ,\ \forall i \in \mathbb{N} \ , \ \mbox{exist} \ \displaystyle \lim_{i \to \infty}(a_i)\}$$
$$l^{\infty}=\{ (a_i)_{i \in \mathbb{N}} \ ...
0
votes
1answer
33 views
How to expand equation inside the L2-norm?
I want expand an L2-norm with some matrix operation inside.
Assume I have a regression $Y=X\beta+\epsilon$.
I want to solve (meaning expand),
$$\displaystyle\|Y-X\beta \|_{2}^2$$
Should I do:
1)
...
1
vote
2answers
35 views
Function Spaces
What is exactly the difference between $L^2$ space and ${\ell}^1$ space? I believe that one of them is the space of square of square integrable functions.
Does it have to do with one is for series ...
1
vote
1answer
41 views
Existance of function of norm one on normed spaces
On a past exam in a course on functional analysis the following problem is given:
Let $X,Y$ be normed spaces and let $x\in X$ be nonzero. Show that there exist some $f\in X^\ast$ s.t. $f(x) = \|x\|$ ...
1
vote
1answer
25 views
Good source for Triebel-Lizorkin spaces?
I'm trying to look into different types of function spaces. At the moment, at least for function spaces involving integration, I only have $L^p$ and $W^{k,p}$. The next function spaces I thought I'd ...
2
votes
1answer
58 views
Inequality for norms
Let g(x, y) be function on $X\times Y$. Show that for all $p\geq q$
$$
\|\,\|g\|_{L^q(Y)}\,\|_{L^p(X)}\leq \|\,\|g\|_{L^p(X)}\,\|_{L^q(Y)}
$$
Thsnk you.
2
votes
2answers
41 views
Equivalent norms and isometries
Let $X$ be a vector space, $||\cdot||_1$ and $||\cdot||_2$ two equivalent norms on $X$. Under what further assumptions can we prove there is an isometry between $(X,||\cdot||_1)$ and ...
7
votes
2answers
90 views
$C[0,1]$ is complete w.r.t. which norm(s)
$C[0,1]$ is complete w.r.t. which norm(s)
$\displaystyle\|f\|_\infty=\sup_{t\in[0,1]}|f(t)|$
$\displaystyle\|f\|_1=\int_0^1|f(t)| \, dt$
$\displaystyle\|f\|_\infty^{0,1}=\|f\|_\infty+|f(0)|+|f(1)|$
...
1
vote
1answer
38 views
Where did I go wrong in showing $\|x\|_1>\sqrt n\|x\|_2$
To show $\|x\|_1>\sqrt n\|x\|_2$ where the norm $\|\cdot||_1,\|\cdot\|_2$ are defined over $\mathbb R^n$
Let $x=(x_1,x_2,\ldots,x_n),y=(1,1,\ldots,1)\in\mathbb R^n$
Using Hölder's Inequality
...
1
vote
1answer
33 views
How to verify whether $(C_{00},\|\cdot\|_p)$ is complete
How to verify whether $C_{00}=\{(x_n):\text{All but finitely many terms are }0\}$ is complete with respect to $p$-norm given by $$\|(x_n)\|_p=\left(\sum_{n=1}^\infty|x_n|^p\right)^{1/p}$$ where $1\le ...
1
vote
1answer
39 views
Relation between Normed space and inner product
Below is what i have proved:
If $V$ is a normed vector space over $\mathbb{R}$ satisfies parallelogram equality, then there exists an inner product $\langle \bullet,\bullet\rangle$ such that ...
2
votes
3answers
30 views
difference between normed linear space and inner product space
I've seen that the definitions of normed linear space and inner product space for a complex vector space $V$ are very close to each other except for the fact that one is defined on $V$ and the other ...
0
votes
0answers
16 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 ...
5
votes
2answers
76 views
Is this a correct question?
This is an exam question in functional analysis which for me doesn't make sense the way it is written. I am asking you if you agree with me on the modifications that needs to be done in the question.
...
2
votes
1answer
56 views
Space of finite dimensional subspaces is separable
In Bernard Maurey's paper "A Note on Gowers' Dichotomy Theorem" at the top of the 7th page, the following fact is stated that I'm not able to prove:
Let $X$ an infinite dimensional separable normed ...
1
vote
1answer
65 views
Is the metric induced by a norm ''unique''?
Let $(X,\rVert{\cdot}\lVert)$ be a normed vector space. Clearly there is a canonical way to induce a metric (and so a topology) on $X$ by defining $d(x,y)=\rVert x-y\rVert$.
Are there other metrics ...
0
votes
1answer
42 views
Operator Norm of a Linear Transformation
PROBLEM For the linear transformation $T:\mathbb{R}^n\rightarrow\mathbb{R}^m$ equipped with the $l^1$-norm, namely, for $x\in\mathbb{R}^n$, $||x||=\sum_{j=1}^n |x_j|$ and similarly for ...
2
votes
2answers
34 views
Picard iterations of a matrix
I need help with this problem. I think i got the first three questions of the exercise, but i'm stuck at the fourth one.
We consider the map $T:{\mathbb{R}^2}\longrightarrow{\mathbb{R}^2}$ defined ...
1
vote
1answer
42 views
Why $ C_{00}$ is not complete with respect to $\sup$ norm?
If $$C_{00}:=\{ x=\{x_n\} \in \mathbb{R^\mathbb{N}}: x_n=0, \forall n>k \text{depending on }x\}$$Can you help me to give such a cauchy sequence in $x$ such that does not converge to $C_{00}$.
3
votes
3answers
55 views
Example of two norms on same space, non-equivalent, with one dominating the other
I am looking for an example (with proof) of two norms defined on the same vector space, such that the norms on the two spaces are NOT equivalent, but such that one norm dominates the other...
4
votes
1answer
49 views
Normed vector spaces and Banach spaces
Let $X$ be a Banach space with norm $||.||$ and let $S$ be a non-empty subset in $X$. Let $F_b(S,X)$ be the vector space of $F(S,X)$ of all functions $f:S \rightarrow X$ such that $\{||f(s)||:s \in ...
0
votes
1answer
37 views
Convergence in normed spaces
I consider a sequence of elements $\{f_n\}_n$ in a normed space $X$ such as $\Vert f_n-f\Vert_X\to 0, n\to\infty$. Let $\{Q_n\}_n$ a complex sequence with $Q_n\to Q$ as $n\to\infty$.
Does $Q_nf_n$ ...
3
votes
2answers
60 views
Banach spaces and quotient space
Let $X$ be a normed vector space, $M$ a closed subspace of $X$ such that $M$ and $X/M$ are Banach spaces.
Any hint to prove that $X$ must be a Banach space?
1
vote
2answers
39 views
Linear functional $\mathscr{L}(E,F)$
Let $\mathscr{L}(E,F)$ denote the space of all linear functionals from $E \to F$.
Let $\mathscr{C}(E,F)$ denote the space of continuous linear functionals from $E \to F$. My question:
How to prove ...
0
votes
0answers
48 views
a question on complete metrizable spaces
There is a claim:
Let $Y$ be a complete metrizable space. If $Y$ is bounded, i.e., $d(Y) < \infty$, then $ \exists C(X,Y)$ is a complete metrizable space.
Why here $Y$ need be bounded?
...
7
votes
1answer
58 views
How does $\lim A_n$ being not invertible imply $\sup_n\|A_n^{-1}\|=\infty$?
Consider a sequence of operators $\{A_n\}_{n=1}^{\infty}\subset B(X,Y)$, where $X,Y$ are normed vector spaces and $B(X,Y)$ denotes the space of bounded linear operators from $X$ to $Y$. Assume that ...
1
vote
1answer
43 views
Study the equivalence of these norms
I have two Hilbert spaces $H_1$ and $H_2$ and I consider a set of functions $f$ which decompose as $f=g+h$ with $g\in H_1$ and $h\in H_2$. I know that this decomposition is unique.
So I define the ...
4
votes
1answer
60 views
If $U$ is finite dimensional, then operator norm is finite
Let $M:U\to V$ be a linear map between normed vector space $U$ and $V$. We know $U$ is finite dimensional (but don't know about $V$). Define $\|M\| = \sup \{\|Mv\|\;:\;\|v\| = 1\}$. I want to show ...
0
votes
2answers
49 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
0answers
45 views
weakly convergent sequence in $l^1$ [duplicate]
Prove that every weakly convergent sequence in $l^1$ converges.
By Riesz on $L^p$ spaces, every linear functional $L\in (l^1 )^*$, is $L(x) = \langle u,x\rangle$ for some $u\in l^{\infty}$ ...
1
vote
3answers
65 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}$$
Note that $x$ is a vector in ...
0
votes
1answer
39 views
Prove that for every positive integer $d$ there exists $C(d)>0$ such that
for every polynomial $p(x)$ with degree $\leq d$, $\max\limits_{x\in[0,1]}|p'(x)| \leq C(d)\max\limits_{x\in [0,1]} |p(x)|$.
There was also a hint given, that says to "use the compactness of a subset ...
0
votes
1answer
58 views
Linear operator
Is there a linear bounded (continuous) operator T from $c$ (convergent sequences with sup norm) ONTO $l^1$ (with its usual norm)?
If it were so (which seems not), using the open mapping theorem we ...
2
votes
1answer
84 views
Prove that if $A, B $ are convex , $B$ is closed, $C$ is bounded and $A+C \subset B+C$ then $A\subset B$
Given the sets $A,B,C \in \mathbb{R}^n$ such that:
$$A+C \subset B+C$$
Show that if $A,B$ are convex, $B$ is closed and $C$ is bounded then $A\subset B$.
I kind of understand the geometrical ...
2
votes
0answers
60 views
Which are nontrivial examples of analytical functions on Frechet spaces?
Let $X$ be a real linear topological space, which is a (separable) Frechet space, such that the topology on $X$ is generated by the countable family $\{p_n:n\in\omega\}$ of norms . A real-valued ...
2
votes
0answers
22 views
Are these two definitions for dual norm equivalent
Suppose there is a p norm, also is a function $\left\|x\right\|_p$, the dual function at $z$ of it is,
$$\sup\limits_x\left\{ z^{T} x-\left\|x\right\|_p\right\}$$
The second is,
$$
\sup\limits_x ...
-1
votes
1answer
71 views
Is $(\ell^1 , \| \cdot \| )$ a Norm space?
Suppose $ x= \{x_n \} \in \ell^ 1$ and $\| x \| = \sup | \sum_{k=1}^n x_k | $, let $ \|x\|_1 = \sum_{n=1}^{\infty} |x_n |$ is a norm for $ \ell^1 $ . Is $(\ell^ 1 , \| \cdot \| )$ a Normed ...
5
votes
1answer
82 views
Is $(l^1 ,\|.\|)$ a Banach space?
Suppose $x=\{x_n\}\in l^1$ and $\|x\|=\sup|\sum_{k=1}^{n}x_k|$, let $\|x\|_1=\sum_{n=1}^{\infty}|x_n|$ is a norm for $l^1$ . Is $(l^1 ,\|.\|)$ a Banach space?
1
vote
1answer
30 views
Proving $\ell_\infty$ is complete
I start by taking a Cauchy sequence $(a_i)$ in $\ell_\infty$. I denote the terms of $(a_i)$ as $f_1, f_2, f_3, \dots$ and so on.
For each $x \in \mathbb{N}$, the sequence $(f_1(x), f_2(x), f_3(x), ...
4
votes
1answer
113 views
Existence of a non zero element in the dual
Let $S$ a vector subspace of a normed vector space $X$ such that $\overline{S} \neq X$. Show that, with the Hahn-Banach Theorem (Geometric Version), that there is $F\in X^{\prime}$ such that ...
2
votes
1answer
33 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, ...
2
votes
2answers
103 views
If $x\mapsto \| x\|^2$ is uniformly continuous on $E$, the union of all open balls of radius $r$ contained in $E$ is bounded $\forall r > 0$
A subset $E$ contained in $\mathbb{R}^n$ is such that the function $x \mapsto \left\Vert x\right\Vert^2$ is uniformly continuous on $E$. For $r > 0$, let $E_r$ denote the union of all open balls ...
3
votes
1answer
48 views
How to show $A=\{T\in \mathcal{L}(\mathbb R^m, \mathbb R^n); \textrm{T is onto}\}$ is open in $\mathcal{L}(\mathbb R^m, \mathbb R^n)$
How can we show $A=\{T\in \mathcal{L}(\mathbb R^m, \mathbb R^n); \textrm{T is onto}\}$ is open in $\mathcal{L}(\mathbb R^m, \mathbb R^n)$?
Here $\mathcal{L}(\mathbb R^m, \mathbb R^n)$ is the set of ...
1
vote
2answers
80 views
$l_1$ equipped with the sup norm is NOT a Banach Space
Prove that $l_1 = \{ x = (x_k)_{k\in\mathbb{N}}\subset \mathbb{R};\ \sum_{k\in\mathbb{N}}\ |x_k| < +\infty \}$ equipped with the norm
$\| x\| = \mathrm{sup}_{k\in\mathbb{N}} |x_k|$ is NOT a Banach ...
2
votes
0answers
29 views
Finding an orthornormal basis given a bilinear form
Take $ \langle \cdot,\cdot\rangle$ to be a nondegenerate symmetric bilinear form (which may not be positive definite). Take $\Vert x\Vert = \vert \langle x,x\rangle \vert^{1/2}$. I want to show that ...
