0
votes
1answer
69 views

Separability of a normed space

The following is an Exercise 12, page 75 of conway's Functional Analysis. Let $\oplus_\infty X_i = \{x\in \sqcap X_i: ||x||=\sup||x(i)||<\infty\} $ where each $X_i$ is a normed space for $i\in I$. ...
3
votes
1answer
54 views

Show that a normed Vector space is complete, need smart help.

I want to show that a normed vector space is complete. I know that if you can show that every Cauchy sequence converges, then it is complete. But in a normed vector space, completeness is equivavlent ...
1
vote
0answers
42 views

The Hilbert space $\mathcal{H}_\eta$ and unitary correspondence with $L^2[a,b]$

The question I have is related to a problem in Stein and Shakarchi's Real Analysis, Chapter 4. The problem Let $\eta(t)$ be a fixed strictly positive continuous function $[a,b]$. Define ...
1
vote
2answers
47 views

a functional analysis question

$X$ is a banach space and $f$ a non zero linear functional. I'm trying to show $null(f)$ not dense in X $\implies f$ continuous. I've tried a few approaches but I think the following seems the most ...
0
votes
1answer
37 views

Continuous linear functional

I want to show that $f:(\ell^1,\parallel. \parallel_1)\to \mathbb K$ defined by $f((x_n))=\sum\limits_{n=1}^{\infty}\dfrac{\vert x_n\vert}{n}$ is continuous linear functional and the norm of $f$ is ...
0
votes
1answer
16 views

Range of a continuous linear mapping

I want to show that the range of the linear map $T:(\ell^1,\parallel .\parallel_1)\to (\ell^2,\parallel .\parallel_2)$ defined by $Tx=x$ is not closed. I considered a sequence $(x^{(n)})$ in ...
1
vote
2answers
116 views

Any two norms on finite dimensional space are equivalent

Any two norms on a finite dimensional linear space are equivalent. Suppose not, and that $||\cdot||$ is a norm such that for any other norm $||\cdot||'$ and any constant $C$, $C||x||'<||x||$ for ...
0
votes
1answer
134 views

$X$ normed linear space separable $\Longleftrightarrow$ $\exists K \subset X$ compact s.t. $\overline{ \text{span}\{K\}}= X$

Let $X$ be a normed linear space. Show that $X$ is separable if and only if there is a compact subset $K$ of $X$ for which $\overline{ \text{span}\{K\}}= X$ I can't figure out how to solve this ...
0
votes
1answer
26 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 ...
2
votes
1answer
50 views

Existence of bounded linear operator with kernel reduced to $\{0\}$

If $X$ and $Y$ are normed spaces, why there must exist a bounded linear operator $T$ from $X$ to $Y$ such that $T(x)$ is not equal to $0$ for all non-zero $x$?
2
votes
0answers
154 views

Prove: Completion of a normed space is a Banach space.

Me again with another (most likely) easy problem that is too hard for me as of now. Suppose we have a normed space $(X,||\cdot||)$ and a map $i:X\rightarrow \hat X$ with $\overline{i(X)} = \hat ...
1
vote
1answer
31 views

Let V be a normed vector space. Show that its norm is induced by a scalar product if and only if it satisfies the parallelogramm inequality.

I managed to prove left to right, but I found it hard to get the other direction even if I have the polarization identity. I've found Fr├ęchet - Von Neumann - Jordan theorem that proves exactly what ...
0
votes
1answer
34 views

Help in proving that a vector norm satisfies an axiom.

I am trying to prove if the following is a vector norm: ||x|| = max{$|x_1 + x_2|, |x_2 + x_3|, |x_3 + x_1$|} (x is vector with 3 elements) I'm stuck proving that $||\alpha x||=|\alpha|*||x||$. I ...
-3
votes
1answer
55 views

Conditional number: exercise

Let's say we've got a $202 \times 202$ matrix $A$ for which $||A||_2=100$ and $||A||_F=101$ (the Frobenius norm). How can we find the sharpest bound (lower) on the 2-norm condition number of $A$? ...
1
vote
1answer
59 views

Series constructed from a cauchy sequence

Given a cauchy-sequence $\{x_i\}_{i\in \mathbb{N}}$ in a normed space $X.$ I need to construct a series that converges in $\mathbb{R}$ with $\{y_i\}_{i\in \mathbb{N}}$ a sequence in $X$: ...
0
votes
1answer
96 views

a normed vector space is normed closed iff it is weakly closed.

The claim is A subspace of a normed vector space is normed closed iff it is weakly closed. I can show one direction. Strong convergence implies weak convergence, so it is weakly closed. But I have ...
0
votes
1answer
92 views

normed vector space real analysis

Prove that $\lVert x\rVert = \left(\sum_{k\in\mathbb{N}} \lvert x_k\rvert^p\right)^{1/p}$ is not norm for $\ell^p = \{x = (x_k)_{k\in \mathbb{N}} : \sum_{k\in\mathbb{N}} \lvert x_k\rvert^p < ...
2
votes
1answer
156 views

Prove that two norms are equivalents

Two norms $\|\bullet\|_1$ and $\|\bullet\|_2$ are equivalents iff $\;\exists\;c_1,c_2>0$ such that $c_1\|x\|_1\le \|x\|_2\le c_2\|x\|_1$ We're working in $\mathcal C^1[0,1]$, and I have ...
0
votes
0answers
38 views

Norm of a mapping

$$C[0,1]=\{f:[0,1]\rightarrow R | \text{$f$ is continuous function}\}$$ $$\|f\|=\max \{|f(x)|:x\in [0,1]\}$$ $$A:(C[0,1],\|\|)\rightarrow(C[0,1],\|\|)$$ $$A(f)(x)=(x^4-x^2)f(x)$$ I have to ...
1
vote
0answers
117 views

Proof that normed space is Banach space

I have to prove that $(l^{\infty},\|\cdot\|_{\infty})$ is Banach space and I have some difficulties. This is what I've done. $l^\infty=\{x=\langle x_k\rangle, k\in N|\exists M>0 \ such\ ...
0
votes
1answer
89 views

Showing a norm preserving isomorphism of vector spaces

Lets define $l^1$ as the complex vector space of all absolutely summable sequences of complex numbers and and $x_0$ consists of all the sequences in $l^\infty$ (all bounded sequences) that eventually ...
1
vote
2answers
56 views

how to show that $A_kB_k\to AB?$

Let in the space $M(n,\mathbb R)=$ set of all $n\times n$ real matrices endowned with $\| \cdot \|_2,~A_k\to A,~B_k\to B.$ Then how to show that $A_kB_k\to AB?$
4
votes
1answer
275 views

Does there exist such a closed subspace of normed linear space

let $(X,|| || )$ be a norm linear space. And $M$ be a closed subspace of norm linear space .does there exist a closed subspace $N$ such that $X=M \oplus N $ . I know such an subspace $N$ exist .but i ...
1
vote
1answer
54 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. $f(x) = \|x\|$ ...
2
votes
1answer
122 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.
0
votes
1answer
260 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
104 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 ...
4
votes
1answer
135 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 ...
3
votes
2answers
249 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?
3
votes
1answer
112 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 ...
4
votes
1answer
143 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 ...
3
votes
2answers
205 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 ...
4
votes
0answers
69 views

Find Lipshitz constant

Let $x, y$ be two vectors in $R^n$. Let $\pi$ be permutation on $\{1, \ldots, n\}$ with uniform distribution. Find Lipshitz constaants of the following functions: $$ f(\pi)=\sum_{i=1}^nx_{\pi(i)}y_i ...
1
vote
1answer
48 views

Does $\|z\|=(\sqrt{|x|}+\sqrt{|y|})^{\frac{1}{2}}$ define a norm on $\mathbb R^2$

Does $\| z\|=(\sqrt{|x|}+\sqrt{|y|})^{\frac{1}{2}}$, with $z=(x,y)\in\mathbb R^2$, define a norm on $\mathbb R^2$?
2
votes
1answer
83 views

A specific linear operator between Banach spaces

Let B be the Banach space $B=(C[0,1],\|\cdot\|_{\infty}$) and let $\{\xi_i\}\in l^\infty$. Let $T:l^1\rightarrow B$ be the linear operator given by: $(Ta)(x) = \sum_n\xi_na_nx^n$. I have three ...
6
votes
2answers
319 views

equivalence of norms

I would like a little help here: I have two defined norms over $C^{1}([0,1])$ : $\| A(f)\|=|f(0)|+\max_{x\in[0,1]}{|f'(x)|}$ $\| B(f)\|=\int_0^1|f(x)|dx+\max_{x\in[0,1]}{|f'(x)|}$ I already ...
3
votes
1answer
142 views

convergence of sequence of averages the other way arround

In a vector normed space, if $ \{x_n\} \longrightarrow x $ then $ z_n = \frac{x_1 + ...+x_n}{n} \longrightarrow x $ Is it true the other way arround too? meaning: if $ z_n = \frac{x_1 + ...
1
vote
1answer
93 views

Regarding an isomorphism between a subspace of $\ell^{\infty}$ and $\ell^1$

Let $c_0$ be the subspace of $\ell^\infty$ consisting of sequences that converge to $0$. Show that $c_0$ is a closed subspace of $\ell^{\infty}$ whose dual space is isomorphic to $\ell^1$. Conclude ...
0
votes
1answer
89 views

Distance of point for a set in linear spaces

Let $X$ a normed linear spaces, $Y \subset X$ a subspace and $z \in X$ an arbitrary point. How can we show that: $$\text{dist} (z, Y) = \sup \{\psi(z) \ | \ \|\psi\| = 1, \psi \equiv 0 \ \text{on} \ ...
1
vote
2answers
79 views

Inequality regarding weak-* convergence

Let $X$ be a normed linear space, $\psi \in X^{*}$ and $\displaystyle \{\psi_n\}_{n \in \Bbb N}$ a sequence in $X^{*}$. Show that if $\displaystyle \{\psi_n\}_{n \in \Bbb N}$ converges weak-${*}$ to ...
1
vote
1answer
37 views

Maximun norm over the complex sequence

Is $C_0$ (the space of all the complex sequences that satisfy $\lim_{n\rightarrow \infty }{x_n} =0$ ) is a Banach space relative to the maximum norm ( $\|x\| =max|x_n| $) and pairwise operations ? ...
1
vote
2answers
63 views

There does not exist $c \ge 0$ such that $\|f\|_{\max} \le c\|f\|_1$

Let $f \in C[a,b]$ and let $\|f\|_1$ be the $\mathcal{L}^{1}$-norm and $\|f\|_{\max} = \max_{x \in [a,b]}|f|$. They are both norms on the given vector space. I want to prove that $\not \exists c \ge ...
2
votes
1answer
118 views

Equivalent statements of continuity of linear operators

I am asked to prove that the following are true: Given a linear operator $T: X \to Y$ where $X,Y$ normed linear spaces: (1) $T$ continuous at at point $\iff$ $T$ continuous everywhere (2) $T$ ...
4
votes
2answers
108 views

Normed linear space and linear functional

Let $X$ be the normed linear spaceof sequences of reals that have only finitely many non-zero terms. Given $x = \{x_n\} \in X$, define $$f(x) = \displaystyle \sum_{n=1}^{\infty} x_n$$ I think that it ...
2
votes
0answers
108 views

$\ell^0$ and $\ell^{\infty}$ norms

Let $x \in S^{n-1}$ and such that its coordinates $|x_1|\geq \cdots \geq |x_n|$. Under which condition on $\|x\|_0$ the following inequality is true that $$\|x\|_{\infty}\leq \frac{1}{\sqrt ...
2
votes
2answers
168 views

equivalence of norms and direct sum

Let $(X,\|\cdot\|_X) $ be an infinite dimensional Banach space. Suppose that you can write $X=V\oplus W$. Write $x=v+w$ and define $(V\oplus W,\|x\|_\oplus :=\|v\|_X+\|w\|_X)$. Show that ...
8
votes
1answer
198 views

A commutator identity for bounded linear maps and the identity operator of a non-zero normed space is never a commutator

Let $ \mathcal{X} $ be a normed linear space and $ S,T: \mathcal{X} \to \mathcal{X} $ be linear operators such that $ S \circ T- T \circ S=1 $. Show that $ S \circ T^{n+1}- T^{n+1} \circ S=(n+1)T^n ...
2
votes
1answer
149 views

Equivalence of two norms

Define two norms as following: $$ \left\Vert f\right\Vert _{1}={ \max_{0\leq x\leq1}\left|f\left(x\right)\right|} , \quad\text{ and }\quad \left\Vert f\right\Vert ...
1
vote
3answers
146 views

Show that a linear functional does not belong to a dual space.

So I have the following statement to prove Let $L:C\:[0,1]\to \mathbb{C}$ be a linear functional defined by $$ Lf=f(0)$$ Show that $L\notin(C[0,1],||\cdot||_2)^*$, where $||\cdot||_2$ is the ...
2
votes
1answer
430 views

proving $L^\infty$ norm inequality (disprove $\Vert f\Vert_\infty\le\sqrt{n}$)

There are three parts in this question, I've done the first two but not sure about the third one. Also see $L^2$ norm inequality. In the third part, I am asked to show that if $W$ is a ...