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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7 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 ...
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1answer
30 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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12 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
28 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
21 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
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 ...
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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 ...
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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 ...
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3answers
54 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 ...
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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 ...
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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.
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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 ...
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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 ...
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2answers
33 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
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 ...
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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.
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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?
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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 ...
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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 ...
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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 ...
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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?
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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 ...
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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 ...
2
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2answers
37 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 $k > 0$. ...
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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 ...
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1answer
108 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
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 ...
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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 ...
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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 ...
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1answer
32 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. $$\sum_{k=1}^\infty|a_k|^q$$ ...
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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
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0answers
41 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 ...
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2answers
86 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?
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1answer
100 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?
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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 ...
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2answers
30 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 ...
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1answer
36 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 ...
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32 views

Is weakly continuous implies strong continuous?

Let $(X, \|\cdot\|_X), (Y, \|\cdot\|_Y)$ normed spaces, operator $A: X \longrightarrow Y$ weakly continuous. Is $A$ continuous with respect to the strong topology? Thanks in advance for your ideas.
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51 views

How did Von Neumann think of the formula for scalar product?

My question is on the idea behind Von Neumann's formula for the scalar product induced from a norm that satisfies the parallelogram law. $\langle x,y\rangle=\frac 14(\|x+y\|^2-\|x-y\|^2)$. Was it by ...
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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} ...
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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 ...
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1answer
20 views

Example of absolute convergent series divergent [duplicate]

I have learnt that in complete norm space any series that is absolute convergent is convergent. However, I am wondering is there any example of divergent series which is absolute convergent in that ...
3
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2answers
140 views

Can all vector spaces be made into normed spaces?

Can all vector spaces be made into normed spaces (even trivial ones)? Vectorspace could be of infinite dimension. Update: I don't know how to make this question more specific. I am talking about a ...
3
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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 ...
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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 ...
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0answers
26 views

$X$ is reflexive iff $X_R$ is reflexive

Problem: Let $X$ be a complex Banach space, $X_R$ its real version. Show that $X$ is reflexive if and only if $X_R$ is reflexive. My run at the solution: I suppose I should use the theorem, that ...
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1answer
41 views

To find the norm of a linear functional

Let $y\in C[a,b]$ and $f(x)=\int\limits_a^bx(t)y(t)dt$ for all $x\in C[a,b]$. I want to show that $f$ is bounded and $\|f\|=\int\limits_a^b|y(t)|dt$. I tried the problem as follows: ...
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1answer
75 views

To show $T$ bounded

Let $X,Y$ be normed linear spaces and let $T:X\to Y$ be a linear map such that for every absolutely convergent series $\sum\limits_{n=1}^{\infty}x_n$, the series $\sum\limits_{n=1}^{\infty}Tx_n$ ...
0
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0answers
33 views

Prove that $f(x,y)$ defines an inner product [duplicate]

Let $(E,\left\lVert . \right\rVert)$ be a normed vector space defined on $\mathbb{R}$ . We suppose that the norm satisfies the Parallelogram law. Prove that: $$f(x,y)=(1/4)[(\left\lVert x+y ...
2
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1answer
41 views

Using Nash inequality to derive an inequality (from proof in paper)

We work on a domain $\Omega \subseteq \mathbb{R}^N$ with the Dirichlet Laplacian. Let $\lVert \cdot \rVert_p$ denote the $L^p$ norm. I am trying to understand why the following inequality is true: ...