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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1answer
66 views

Sequence of partial sums converge

Let $(X,\|.\|)$ be a Banach space and a sequence $(x_n) \subseteq X$ be such that the series $\sum_{n=1}^{\infty} \|x_n\|$ converges in $\mathbb R$. Prove that the sequence $S_n=x_1+...+x_n$ converges ...
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3answers
360 views

What points is the norm Frechet differentiable at

I know the definition of Frechet derivatives - there exists a bounded linear map... Maybe someone could show me a similar example on how to approach questions like these.
5
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1answer
51 views

Does there exist an uncountable dimensional real vector space $X$ such that $(X,\|\cdot\|)$ is a Banach space for any norm $\|\cdot\|$ on it?

Does there exist an uncountable dimensional real vector space $X$ such that $(X,\|\cdot\|)$ is a complete space for any norm $\|\cdot\|$ on it ?
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1answer
47 views

Proving these norms are not equivalent

$X$ is the vector space containting all polynomials with real coefficients. For every $P \in X$, define $N_1(P)= \sup _ {t \in [0,1]} |P(t)|$ and $N(P)=N_1(P)+|P'(1)|$. Prove that $N$ is not ...
5
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1answer
71 views

Can any uncountable dimensional real vector space be made into a Banach space?

On any real vector space $V$ of uncountable dimension , can we always define a norm such that endowed with that norm , $V$ becomes a complete normed linear space ? ( I know it can be done if $V$ is ...
2
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1answer
64 views

$X$ be a real normed linear space ; if $\mathcal L(X,X)$ is complete then is $X$ also complete?

Let $X$ be a real normed linear space and $\mathcal L(X,X)$ denote the set of all bounded linear operators on $X$ , we know that if $X$ is complete then so is $\mathcal L(X,X)$ ; is the converse true ...
2
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1answer
86 views

Lipschitz map between metric and normed spaces

Let be $F:(X,d)\to V$ a map between $(X,d)$ metric space and $V$ normed space, such that for each $f\in V'$ (linear and continuous), $f\circ F$ is lipschitz map. Show that $F$ is a Lipschitz map. I ...
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0answers
44 views

How to see if a norm is finite

How do you know if $$||\frac{1}{\sqrt3},\frac{1}{\sqrt8}, ... , \frac{1}{\sqrt{n^2-1}}, ... ||_2$$ is finite. So this means is $$\bigg( \sum _{n=2}^{\infty} | \frac{1}{\sqrt{n^2-1}}|^2 \bigg) ^{\...
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0answers
6 views

Two points in a polygonal-path-connected set can be connected with a non-intersecting polygonal path

Let $X$ be polygonal-path-connected and $x,y\in X$. So $x$ and $y$ can be connected by a polygonal path $P=\bigcup_{i=1}^n L_i$ where $L_i$ is a line segment $[x_i,x_{i+1}]$. Non-intersecting means ...
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1answer
36 views

Show that $ f(x)=\sum_{n=1}^{\infty} 2^{-n} f_n(x)$ defines a continuous function on $(0,\infty)$

Let $f_n$ be a sequence of continuous functions on $(0,\infty)$ with $|f_n(x)|\le n$ for every $ x>0$ and $n\ge1$, and such that $\lim_{x\to\infty} f_n(x) =0$ for each $n$.Show that $ f(x)=\sum_{n=...
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1answer
360 views

Frechet Derivatives of normed spaces

(a) Would I use the definition of an open set for one U? How do I show the function is Frechet differentiable. I know the definition but not sure how to apply it. $\lim_{h\to 0}\frac{\lVert f(x+h)-f(...
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1answer
44 views

Proving polynomial v.s. is a norm

Let $X$ be the vector space containing all polynomials with real coefficients. For every $P ∈ X$, define $N_1(P) = \sup_{t∈[0,1]} |P(t)|$ and $N(P) = N_1(P) + |P'(1)|$. I have to prove $N_1$ is a norm ...
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1answer
576 views

For every normed space the norm map is not Fréchet differentiable at $0$.

Argue that for every normed space $\mathbb{X} \neq \{ 0 \}$ the norm map $\| \ldotp \|_\mathbb{X} : \mathbb{X} \to \mathbb{R}$ is not Fréchet differentiable at $0$. Not really sure where to start on ...
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2answers
58 views

Prove that $C^1[0,1]$ is space of continuously differentaible function with $C_1$ norm is separable.

$C^1[0,1]$ is space of continuously differentiable function with $C_1$ norm.Then the space $ (C^1[0, 1],)$ is a separable space. I am thinking of c^1[0,1] is subset of c[0,1], and c[0,1] is separable. ...
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1answer
57 views

Proving something is a Banach Space

Prove that $(\ell ^∞,||·||_∞)$ is a Banach space using the following steps. Let $(x_n)_{n∈\mathbb N}$ be a Cauchy sequence in $(\ell ^∞,||·||_∞)$. For $n > 1$, let $x_n = (a_{1,n},a_{2,n},...,a_{...
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0answers
14 views

prove that a non constant linear functional from a normed linear space X is discontinuous if and only if Z(f) = {x in X | f(x) = 0} is dense in X. [duplicate]

I could prove that if Z(f) is dense in X then the functional must me discontinuous but I am not able to prove the other way round
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1answer
119 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
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1answer
102 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 ...
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0answers
24 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 ...
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1answer
98 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 ...
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1answer
26 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) ...
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1answer
36 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 ...
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2answers
34 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 $\|h-...
3
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2answers
86 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 ...
4
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3answers
57 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 where $...
1
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1answer
29 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 [0,1]...
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1answer
28 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.
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1answer
21 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 \bar{...
1
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1answer
68 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 $\|f\|=\infty$...
3
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2answers
63 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 \...
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2answers
28 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 ...
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0answers
20 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.
5
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2answers
76 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?
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2answers
145 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
24 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 $(a,z)(b,w)=(ab+zb+wa,...
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1answer
53 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 ...
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0answers
33 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 $A=\{...
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0answers
23 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 $$x^*(x)=\sum_{n\in\mathbb{N}}...
2
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2answers
47 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 $k > 0$. ...
0
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1answer
121 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 ...
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1answer
217 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 ...
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0answers
153 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
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1answer
220 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 ...
0
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1answer
43 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. $$\sum_{k=1}^\infty|a_k|^q$$ ...
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 \delta_q(u,v)=\left(\int_0^...
0
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2answers
89 views

Closure of a subset of a normed space is equal to the set of limits of sequences in $A$

I'm not sure how to show that the closure of a subset $A$ of a normed space $\mathbb{X}$ is equal to the set of limits of sequences in $A$. Could someone help me?
0
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1answer
109 views

Completeness of a finite dimensional linear subspace of X

How can I show that a finite dimensional linear subspace F of an arbitrary normed space X is complete, hence closed?
1
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1answer
40 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 \...
1
vote
2answers
38 views

Proving something is a norm

Let $a<b$ be real numbers and $X=C[a,b]$ be the space of continuous functions $f:[a,b] \rightarrow \mathbb R$ Prove that $||f||_1 = \sup _{t \in [a,b]} |f(t)|$ indeed defines a norm on $X$. The ...
0
votes
1answer
50 views

Operator Norm: Why is this formulation so rarely seen?

There are many equivalent definitions for the norm $||T||$ of a linear operator $T:X\to Y$. Citing the Wikipedia and some books about functional analysis, the equivalence of definitions are often ...