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.

learn more… | top users | synonyms

3
votes
2answers
37 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 ...
1
vote
1answer
15 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 ...
1
vote
1answer
12 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.
0
votes
1answer
15 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 ...
3
votes
2answers
32 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 ...
0
votes
0answers
23 views

If $u \in W^{1,p}(I) \bigcap C_c(I)$ then $u \in W_{0}^{1,p}(I) $ where $ W^{1,p}(I)$ is the Sobolev Space

I want to show the following statement: If $u \in W^{1,p}(I) \cap C_c(I)$ then $u \in W_{0}^{1,p}(I) $. $W^{1,p}(I) $ is the Sobolev Space, i.e. the space consisting of the functions that are ...
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 ...
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.
4
votes
2answers
52 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
2answers
117 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
16 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
62 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 ...
2
votes
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 $k > 0$. ...
-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
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 ...
0
votes
1answer
170 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
0answers
145 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
201 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
votes
1answer
31 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
113 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
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 ...
0
votes
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?
0
votes
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?
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
0answers
31 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.
2
votes
0answers
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 ...
2
votes
0answers
67 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} ...
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 ...
1
vote
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
votes
2answers
138 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
votes
2answers
193 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 ...
10
votes
2answers
176 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 ...
1
vote
0answers
25 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 ...
1
vote
1answer
40 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: ...
4
votes
1answer
74 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
votes
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
votes
1answer
40 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: ...
2
votes
2answers
59 views

Is a Normed Vector Space Necessary to Prove Path Connectedness?

Path connectedness seems to be defined in a topological space, but can the existence of a path be proven without using the functions of vector addition, scalar multiplication and norm ? For example, ...
0
votes
1answer
31 views

How to interpret these two equalities involving 2-norm?

What is the difference between $$\Vert x+y \Vert_2^2 $$ and $$\Vert x+y \Vert_2 $$ Can we write $$\Vert x+y \Vert_2 \stackrel{?}{=} \sqrt{\Vert x+y \Vert_2^2} \tag{*}$$ Moreover, when does (1) ...
2
votes
1answer
35 views

A normed space is Banach iff its unit sphere is complete [duplicate]

Let $X$ be a non-trivial (other than singleton $x$) normed space. Prove that $X$ is a Banach space if and only if $\{x \in X \mid \|x\| = 1 \}$ is complete.
0
votes
0answers
17 views

Does nonexpansive property in H-norm imply nonexpansive in 2-norm?

Suppose $\|f(x) - f(y)\|_H \le \|x - y\|_H$. In other words, $f$ is nonexpansive in the norm with respect to positive definite H: $\|z\|_H = z^T H z$. Can we then say something along these lines: ...
0
votes
1answer
14 views

Norm of pointwise product of Lp functions

Does the following inequality hold in $L_p$ spaces? $\|fg\|_p\leq\|f\|_p\|g\|_p$ How would I go about proving this? Do I need to apply Cauchy Schwarz?
2
votes
2answers
14 views

If $V$ is completely normable, then is every norm complete?

Here is a theorem that motivated my question. Let $(V,||\cdot||_V)$ be a normed space over $\mathbb{K}$. Then, there exists a Banach space $(X,||\cdot||_X)$ such that $V$ is dense in $X$ and ...
0
votes
0answers
25 views

Definition of James' Space

The definiton of James' Space in wikipedia begins with: Let $\mathcal{P}$ denote the family of all finite increasing sequences of integers of odd length. Shouldn't it be all finite increasing ...