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.

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30
votes
5answers
6k views

Not every metric is induced from a norm

I have studied that every normed space $(V, \lVert\cdot \lVert)$ is a metric space with respect to distance function $d(u,v) = \lVert u - v \rVert$, $u,v \in V$. My question is whether every metric ...
0
votes
0answers
15 views

Closure of compact operators?

If $X,Y$ are normed spaces (not nec complete) and $A_n:X\rightarrow{Y}$ bounded linear operators of finite rank converging in the operator norm to $A$ is $A$ compact? A diagonal argument and cauchy ...
0
votes
3answers
88 views

Norm of vectors inequality [duplicate]

I tried proving this with triangular inequality but i was not right can any one help me with this
0
votes
1answer
31 views

Convergent Bounded Linear Maps

I'm not sure how to show that the composition of two convergent bounded linear maps converges to the composition of their limits. I've shown that the composition of bounded linear maps is a bounded ...
1
vote
1answer
40 views

Fréchet derivative of $f(x) = x$

Im not sure how to find the Fréchet derivative of the function $f : \mathbb{X} \to \mathbb{X}$ given by $f(x) = x$, where $\mathbb{X}$ is a normed space. I'm not given the dimension of the normed ...
1
vote
2answers
31 views

Finding distance of $h(t)=t$ from a closed subspace $Y$ of $\pi$-periodic functions in $L^2(-\pi,\pi)$

Let $Y=\{f\in L^2(-\pi,\pi):f(t-\pi)=f(t) \text{for almost all $t\in(0,\pi)$} \}$ Show that there exists $g\in Y$ such that $$\|h-g\|_2=\inf \{\|h-f\|_2:f\in Y\}$$ where $h(t)=t$. Compute ...
1
vote
0answers
13 views

$A,B$ closed subsets of $\mathbb R^n$ , when can we say (other than compact-ness of $A$ or $B$ ) $\exists b \in B$ such that $dist(A,B)=dist(b,A)$ ?

Let $A,B$ be disjoint closed subsets of $\mathbb R^n$ , when can we say ( weaker than compact-ness of $A$ or $B$ ) that there exist $b \in B$ such that $dist(A,B)=dist(b,A)$ ? I know that if $A,B$ are ...
1
vote
1answer
69 views

Two (equivalent ?) norms on Hilbert space

Let $H$ be a vector space equipped with two inner products $\langle \cdot,\cdot\rangle_1, \ \langle \cdot,\cdot\rangle_2$, s.th. $(H,\langle \cdot,\cdot\rangle_j)$ is a Hilbert space for $j=1,2$. ...
0
votes
1answer
30 views

Differentiability of norm

Is a norm from any normed space $\mathbb{A}$ to $\mathbb{R}$ Frechet differentiable at zero? I've tried proving by contradiction that the norm is not Frechet differentiable at zero, but didn't get ...
0
votes
1answer
22 views

Find the value of $||T||$ if T is defined as:

This question was asked in GATE 2016: Please help me to figure out the right answer. Let $T$ ∶ $ℓ_2$ → $ℓ_2$ be defined by $T((x_1,x_2,...,x_n...))$=$(X_2-X_1, X_3-X_2,...,X_{n+1}-x_n,...)$ Then (A) ...
0
votes
1answer
16 views

$X$ is a normed linear space such that for some compact $K\subseteq X$ , $\operatorname{span} K$ is dense in $X$ then is $X$ separable?

Let $X$ be a normed linear space which is separable. Then I know that there exists a compact subset $K$ of $X$ such that $\operatorname{span} K$ is dense in $X$ (in fact we can also find compact and ...
4
votes
3answers
55 views

A non continuous linear map $A:\Bbb{R}[X]\rightarrow \Bbb{R}$ such that $A(P)=P(1).$

I have a linear map $A:\Bbb{R}[X]\mapsto \Bbb{R}$ such that $A(P)=P(1)$, for the $p-$norm : $\Vert P\Vert=\bigl(\sum_{i=1}^n\vert a_i\vert^p\bigr)^{1/p}$ where $p\in[1,+\infty].$ For the cas ...
2
votes
1answer
41 views

Not all norms are equivalent in an infinite-dimensional space

How to prove that not all norms are equivalent in an infinite-dimensional vector space? In particular, I would like to prove that for a space $X$ of continuous real-valued functions defined on ...
2
votes
1answer
40 views

Normed Linear Space - maximum norm vs. $||f||_1$

For $f$ in $C[a,b]$ define $$|| f ||_1 =\int_a^b |f|.$$ a. Show that this is a norm on $C[a,b]$. b. Show that there is no number $c \geq0$ for which $$||f||_{max} \leq c ||f||_1 \ for \ all \ f \ ...
1
vote
1answer
14 views

Infinite dimensional $\sigma$-compact space

It is well-known that Banach space is $\sigma$-compact iff it is finite dimensional. I'm looking for examples of infinite dimensional normed $\sigma$-compact spaces.
1
vote
1answer
17 views

Application of uniform boundedness principle for sequence of operators

Let $T_n:X:=C([0,1])\rightarrow C([0,1])=:Y$ be a sequence of linear operators. Suppose: $T_n$ is bounded $(Tf)(x):=\lim\limits_{n\to\infty}(T_nf)(x)$ exist for every $f\in C([0,1])$ and $x\in ...
3
votes
1answer
46 views

When is the completion of a space of functions a space of functions?

If $V$ is a $\mathbb C$-vector space of functions $f: X \to \mathbb C$ on some common domain $X$ and $\tau$ is a Hausdorff, locally convex topology on $V$, when may the completion of $(V,\tau)$ also ...
4
votes
0answers
49 views
+100

Point set where each point has unity distance to all other points ($L_1$ metric)

I want to construct a point set where each point has the same (w.l.o.g., unit) distance to all other points in the $L_1$ metric. Example: The points $\left(\frac{1}{2},0\right)$, ...
0
votes
1answer
18 views

Proving that a set infers a norm given certain conditions

Let $\bar{B}$ be a set in a vector space $E$. Then $\bar{B}$ is the closed unitary ball for a norm iff $\mathbb{N} . \bar{B} = E$ $\lambda x + (1-\lambda)y \in \bar{B}$ for all $x \, ,y \, \in ...
1
vote
1answer
43 views

If $f$ is not continuous then $\ker f$ is dense in $X$

Let $X$ be a normed space and $f:X\rightarrow \mathbb R$ a linear function. I saw an old post with this problem, but there is not a complete proof. For beginning I have to consider that ...
2
votes
0answers
21 views

Operator norm of matrix of scalars regarded as matrix with entries in the unitization of a Banach algebra

Let $A$ be a non-unital Banach algebra, and let $A^+=\{(a,z):a\in A,z\in\mathbb{C}\}$ with product $(a,z)(b,w)=(ab+zb+wa,za)$ and norm $||(a,z)||=||a||+|z|$. Equip $M_n(A^+)$ with the operator norm by ...
3
votes
2answers
34 views

Composition involving bounded linear operators

I recently come across the following statement mentioned in a proof: Let $X,Y$ be normed linear spaces and $T:X \rightarrow Y$ be a linear operator. if for every bounded linear functional $U: Y ...
3
votes
2answers
199 views

Hahn-Banach Theorem for separable spaces without Zorn's Lemma

I was reading about the Hahn-Banach Theorem, its many versions and their proofs. It's known that in the proofs we need Zorn's Lemma. But in the book that I'm reading, the author said if $X$ is a ...
0
votes
2answers
25 views

Completeness of a certain normed space

let $(X, \| \|$) be the normed linear space of bounded uniformly continuous real valued functions defined on $R$. I need to prove that X is complete (under sup norm). My attempt: Let {$f_n$} be a ...
4
votes
2answers
55 views

Can a continuous map between a Banach space and a non-Banach space be bijective?

Let $\mathbb{X}$ be a normed space that is complete and $\mathbb{Y}$ be another normed space which is not complete. Then can a bounded linear map $A:\mathbb{X} \to \mathbb{Y}$ be bijective or not?
0
votes
0answers
11 views

A linear functional is continuous iff its kernel is closed [duplicate]

A linear functional defined on a normed space is continuous if and only if its kernel is closed.
1
vote
0answers
146 views

Does every Banach space admit a continous injection to a non-closed subspace of another Banach space?

Let $(V, \lVert\,\rVert)$ be a Banach space. I want to produce a non-complete norm $\lVert\,\rVert'$ on it such that $\lVert v\rVert' \leq \lVert v\rVert$ for all $v$ in $V$. Given a continuous ...
5
votes
2answers
2k views

Examples of compact sets that are infinite dimensional and not bounded

In an infinite dimensional Banach space, does a compact subset have to be finite dimensional? I know it cannot contain any infinite dimensional balls, if this mean it has to be finite dimensional, ...
0
votes
2answers
119 views

No normed space such that its dual is equivalent to $C^1[0,1],||,||_{\infty}$

I have showed $C^1$ is not complete by taking $f_n(x)=\sqrt{x+\frac{1}{n}}$ and showed is converges uniformly but limit doe not belong to $C_1$ (not differentiable at 0). Is this correct using the ...
2
votes
0answers
17 views

A condition on quotient norms on quotient Banach algebras

Let $A$ be a non-unital Banach algebra, and let $A^+$ be the unitization of $A$ consisting of elements of the form $(a,z)$ where $a\in A$ and $z\in\mathbb{C}$ with multiplication ...
0
votes
1answer
49 views

Give an example of $X,Y$ and $A$ and a continuous function $f:A \rightarrow Y$ for which there is no continuous extension to $\overline{A}$

Give an example of $X,Y$ and $A$ and a continuous function $f:A \rightarrow Y$ for which there is no continuous extension to $\overline{A}$. $X$ and $Y$ are normed spaces and $\overline{A}$ is the ...
-1
votes
0answers
63 views

Equivalence of normed spaces preserves completeness

I have proved a dual of a normed vector space is complete but the normed vector space is not. Then they cannot be equivalent but why?
0
votes
0answers
30 views

Problem on norm linear spaces

Let $X,Y$ be normed linear spaces and let $T:X\to Y$ be a non-zero bounded linear operator. If $\alpha\geq 0$, I want to prove that $\inf\{\|x\|:x\in X,\|Tx\|=\alpha\}=\frac{\alpha}{\|T\|}$. Let ...
0
votes
0answers
16 views

Dual space and norm to $X=c_0\times l^1$, solution check

$$X=c_0\times l^1,\ \|(x,y)\|=\|x^\|_{\infty}+\|y\|_1$$ It is clear, that the dual space is isomorphic to $l^1\times l^\infty$ and the functional $x^*(x)$ is defined as ...
10
votes
2answers
178 views

Exists a uniformly convex norm on Banach space satisfying certain condition?

Let $E$ be a Banach space with norm $\|\cdot\|$. Assume that there exists on $E$ an equivalent norm, denoted by $|\cdot|$, that is uniformly convex. Given any $k > 1$, does there exist a uniformly ...
2
votes
0answers
71 views

Show that the “folium” is norm closed

A C*-algebra $\mathfrak{A}$ is a Banach algebra with an involution operation $* : \left\lbrace\begin{aligned} \mathfrak{A} &\longrightarrow \mathfrak{A} \\ a &\longmapsto a^* \end{aligned} ...
2
votes
2answers
39 views

If $(x_n)$ is a Cauchy sequence, then it has a subsequence such that $\|x_{n_k} - x_{n_{k-1}}\| < 1/2^k$ [duplicate]

If $X$ is a normed linear space and $(x_n)$ is a Cauchy sequence in $X$, then $x_n$ has a subsequence $x_{n_k}$ which satisfies $$\|x_{n_k} - x_{n_{k-1}}\| < \frac 1 {2^k}$$ for every $k > 0$. ...
6
votes
4answers
382 views

Cauchy sequences - can we control the rate at which elements “get closer”?

In Simon & Reed's book Methods of Modern Mathematical Physics, it is proven in chapter 1 (Theorem 1.12) that $L^1$ is complete (Riesz-Fisher theorem). The proof starts off as follows: Let ...
1
vote
1answer
71 views

A Cauchy sequence has a rapidly Cauchy subsequence

I am trying to fill in the details of a proof related to the Riesz-Fischer Theorem. We need to show that every Cauchy sequence $\{f_n\}$ has a rapidly Cauchy subsequence. My text claims that we can ...
0
votes
1answer
202 views

Dimension and basis of bounded linear maps (products)

Let $X, Y, Z$ be finite dimensional normed spaces with bases $\{x_1,...,x_l\}$, $\{y_1,...,y_m\}$, $\{z_1,...,z_n\}$ respectively. What is the dimension of $\mathcal{L} \{X \times Y; Z\}$ and give ...
-1
votes
0answers
142 views

Continuity of functions and relative closure of preimages

For X and Y, two normed spaces, let E be a subset of X and $f\colon E \to Y$. Show that $f$ is continuous if and only if for every closed set A in Y, it's preimage $f^{-1}(A)$ is relatively closed in ...
0
votes
1answer
109 views

Let A a subset of X be a finite dimensional linear subspace. Show that A is complete

Let X be a normed space. Let A a subset of X be a finite dimensional linear subspace. Show that A is complete (even if X is not). Using the above show that A is a closed subset of X. For the first ...
0
votes
0answers
146 views

Show that there is no normed space X s.t. its dual X' is equivalent to $((C^1[0,1];\mathbb{R}), ||.||_{\infty})$.

Show that there is no normed space X s.t. its dual X' is equivalent to $((C^1[0,1];\mathbb{R}), ||.||_{\infty})$. Normed spaces are equivalent if there are bounded linear maps which are inverses of ...
0
votes
1answer
171 views

Equivalence of dual space of normed space X and continuously differentiable functions.

Define that two normed spaces $X$ and $Y$ are equivalent if there exists bounded linear maps $A: X \to Y$ and $B: Y \to X$ such that $A$ and $B$ are inverses of each other. How do you show that there ...
0
votes
1answer
34 views

Show $\sum_{k=1}^\infty|a_k|^q$ converges [duplicate]

Let $1 \leq p < q < \infty$. Show that if $x=(a_k)∈ℓ^p$ i.e. the condition that the series $$\sum_{k=1}^\infty|a_k|^p$$ converges holds, then $x∈ℓ^q$ i.e. $$\sum_{k=1}^\infty|a_k|^q$$ ...
0
votes
2answers
115 views

Continuity of a function on a normed space

If $\mathbb{X}$ and $\mathbb{Y}$ are normed spaces, and E is a subset of $\mathbb{X}$ such that $f : E \rightarrow \mathbb{Y}$. How can I show that f is continuous if and only if for every closed ...
2
votes
0answers
42 views

Distinct topologies on continuous functions $(L_p$ spaces restricted to $C[0,1])$

I am trying to show that the topologies $\tau_p$, $\tau_q$, $\tau_{\infty}$ ($1\leq p<q$) induced by the metrics $$\delta_p(u,v)=\left(\int_0^1 |u-v|^p\right)^{1/p}\quad ...
0
votes
2answers
29 views

Show that there exist $\{x_n\}$ a sequence of elements in $A- \{x\}$ such that $x_n \to x$

Let $E$ a normed vector space and $A \subset E$. Let $x$ an accumulation point in $A$. Show that there exist $\{x_n\}$ a sequence of elements in $A- \{x\}$ such that $x_n \to x$. Definition : An ...
0
votes
0answers
51 views

If a linear operator is strong-weak continuous, then it is bounded

$X$ and $Y$ are normed spaces and $L: X\to Y$ is a linear operator from $X$ to $Y$. Show that if $L$ is a continuous operator from $X$ with the topology to $Y$ with the weak topology, so $L: X\to ...
1
vote
1answer
35 views

Show that $l^p \subseteq l^q$ for $1 \leq p < q < \infty$

$$l^p = \{ (a_k)_{k \geq 1} : \sum \limits_{k=1}^{\infty} |a_k|^p < \infty \}$$ Since it is said $l^p \subseteq l^q$, I would have thought we have to show $$\sum \limits_{k=1}^{\infty} |a_k|^q ...