Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [normed-spaces]

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.

0
votes
1answer
17 views

A Vector Divided by it's Distance to a Subspace Yields a Vector with Distance 1 to Subspace

Let X be a normed vector space, S a subspace of X and x∈X. Distance is defined by: $$|x,S|:=\inf||x-s||, x\in X, s\in S$$ How does one prove that for: $$z:=\frac{x}{|x,S|}\Rightarrow |z,S| = 1$$ This ...
0
votes
1answer
15 views

p - adic norm not equivalent to usual norm!

I know that the two norms: $p$ - adic norm and the usual norm ($\left| \cdot \right|$) defined on $\mathbb{Q}$ are not quivalent. This is clearly because the $p$ adic norms staisfies the strong ...
1
vote
0answers
30 views

A question regarding Radon-Riesz property

I want to show that if a normed linear space has Radon-Riesz property, then it has the semi Radon-Riesz property. We know that a normed linear space is said to have Radon-Riesz property if for every ...
2
votes
1answer
20 views

Polar set of the open ball in the dual space $X'$

Assume $(X,\Vert\cdot\Vert)$ is a normed space with the dual space $X'$. I want to show that the polar set of the open disk $U_{r}^{X'}(0)$ is equal to the closed disk $K_{\frac{1}{r}}^{X}(0)$, e.i.: $...
1
vote
2answers
25 views

Choosing proper sequence given a limit

Suppose $X$ is a Banach space and $X_j$ are subsets with $X_1 \subseteq X_2 \subseteq X_3 \subseteq \ldots$ so that $X= \overline{\bigcup _{j=1}^{\infty} X_j }$. If $u \in X$ then $u= \lim_{n \...
2
votes
1answer
29 views

Complement a finite dimensional subspace in a Banach space

Given a Banach space $(X,\|,\|)$, and a finite dimensional subspace $F \subset X$, is it always possible to choose a closed linear complement to $F$. Explicitly, I mean to say, will there always exist ...
1
vote
1answer
33 views

Check if $||.||_{1}$ and $||.||_{\infty}$ are strictly convex norms on $\mathbb{C}^n$ for $n \geq 2$.

A norm is strictly convex on a given normed space $X$ if, for every $x,y \text{ with } x \neq y \in X$ of norm 1, we have that $||x + y|| < 2$. $||.||_{1} = \sum_{j}^{n}|x_j|$ Say we take the set ...
1
vote
1answer
34 views

Convergence of trace class operators in Hilbert Schmidt norm

Let $\mathscr{A}_n$ be a sequence of trace-class operators on a Hilbert space $\mathcal{H}$ and let further $\mathscr{A}$ be another trace-class operator on the same space. Assume that $\mathscr{A}_n$...
5
votes
0answers
57 views

How is “continuous dependence on initial conditions” defined? And how to prove it?

EDIT: I tried to prove the continuous dependence of the problem by somehow use a weak Maximum principle and posted a modified Version of this post on mathoverflow: https://mathoverflow.net/questions/...
5
votes
1answer
179 views

Counterexample for “continuous image of closed and bounded is closed and bounded” (in normed spaces).

It's well known that: If $X$ is a finite-dimensional normed space, $C$ is a closed and bounded subset of $X$ and $f:C\subset X\to X$ is continuous, then $f(C)$ is closed and bounded. If $X$ is any ...
3
votes
1answer
56 views

Boundary of Set defined with $\sin(\frac{1}{x})$

Let $S=\{(x,y) \in (0,\infty) \times (-1,\infty) \mid y \geq \sin(\frac{1}{x})\}$ in $(\mathbb{R}^2,\lVert \cdot \rVert_\infty)$. I think that the boundary $\partial S= \{ (0,y) \mid y \in (-1, \infty)...
1
vote
0answers
17 views

Must a subspace of a Euclidean space with zero orthogonal complement be dense?

Let $X$ be a Euclidean space (i.e. a vector space over $\mathbb{R}$ or $\mathbb{C}$ with an inner product, $\textbf{not necessarily complete}$). Let $S$ be a subspace of $X$ with its orthogonal ...
1
vote
1answer
15 views

If a subspace is Banach and Quotient is Banach then the mother space is Banach.

Let $X$ be a Normed Linear Space and $M$ be a closed subspace of X. Assume that both $M$ and $X/M$ are banach spaces. Prove that $X$ is a banach space. So firstly I assumed a Cauchy sequence ${(f_n)}$...
0
votes
1answer
23 views

Extending multiplication to the completion of an algebra

Let $A$ be an algebra with a submultiplicative norm $\|\cdot\|$, that is, assume $$ \|ab\| \leq \|a\|\|b\|, ~~~~~~ \text{ for all } a,b \in A. $$ Let us denote by $\overline{A}$ the completion of $A$ ...
2
votes
1answer
50 views

Hahn-Banach Theorems Applications

Please, if anyone can help me with some useful tips to solve this aim: Let $K^1,...,K^n$ closed convex sets containing the origin of a normed space $E$, and let $c_1,...c_n$ positive real numbers. ...
5
votes
3answers
351 views

Compactness in normed vector spaces.

Let $(X,\Vert\cdot\Vert)$ be a normed $\mathbb{K}$-vector space and $A \subset X$ be closed and bounded. My problem is how to determine whether $A$ is compact? I know that a compact subset is always ...
0
votes
2answers
59 views

Prove that a set is open or closed

Let $M=\{f \in C[0,1] \mid f(0) = 0\}$ with $M \subset (C[0,1],\lVert \cdot \rVert_\infty)$. Is this set open or closed in the given normed vector space? I think it is closed, but I'm not sure how to ...
2
votes
1answer
55 views

Need help to prove statements about linear functional for newbie [closed]

Let $E$ be a normed space and let $ϕ : E → \mathbb K$ be a linear functional $ϕ \neq 0$. Prove the following statements: (i) There exists $x_0 \in E$ with $ϕ(x_0) = 1$. (ii) $E =$ ker$ϕ$ $\oplus$ ...
0
votes
1answer
31 views

$l_p : (p≠2)$ is not an inner product.

I am using Kreyszig's functional analysis it is example in it which says that $l_p : (p≠2)$ is not an inner product space. For proof it uses the standard norm on $l_p$ and shows that it does not ...
2
votes
1answer
16 views

Equivalent operator norm on dense subset

Let $X$ and $Y$ be normed vector spaces and let $X_{0}\subset X$ be a dense subspace. Further, let $T:X\longrightarrow Y$ be a bounded, linear operator. Prove that $||T||_{L(X,Y)}:=\underset{x\in X,\\|...
0
votes
1answer
47 views

Examine whether a linear map is continuous and determine the operator norm

Let $P$ be the space of all real-valued polynomials, defined on $\mathbb{R}$. For a polynomial $p\in P$, such that $p(t)= \sum_{k=0}^na_kt^k$, set $||p||:= \sum_{k=0}^n|a_k|$. Consider the linear ...
0
votes
1answer
30 views

Piecewise function in $L^p$ spaces

Consider the space $C[0,3]$ for piecewise function such that $$f_a(x)= \begin{cases}a^3(2-a^3x),& 0\le x \le \frac2{a^3}\quad\text{and} \\[1ex] 0 , & \...
0
votes
1answer
25 views

Topological an Algebraic properties of the space of continuous function $C(X)$

I am looking for the meaning of topological an algebraic properties of the space of continuous function $C(X)$ where $C(X)$ is a normed vector space with the supremum norm, in general terms (...
0
votes
1answer
19 views

space equipped with quasinorm

I have the following problem. Hope someone can help me. Consider for $N \in \mathbb{N}$ the space $\mathbb{C}^{N}$ of all vectors of length $N$. Now define for $p > 0$ the following norm $$ ||x||_{...
0
votes
1answer
35 views

Questions about deriving the dual space of $l^{1}$

I am an engineering student and I am reading the book "Introductory Functional Analysis " by kreyszig and am lost in the proof of finding the dual space of the $l^{1}$ space . Here is how author ...
1
vote
1answer
40 views

Is it possible that $\forall y \in X \quad \|y-b\| < \max(\|y-a\|, \|y+a\|) $?

Let $(X, \|\cdot\|)$ be a normed space. Let $a, b \in X$ be noncollinear vectors. Is it possible that \begin{equation} \forall y \in X \quad \|y-b\| < \max(\|y-a\|, \|y+a\|)? \end{equation} So ...
0
votes
0answers
28 views

Integral and norm

Consider the fonction : $N(x,y)=\int_{0}^{1}\mid x+ty\mid dt $ where $x,y$ are real numbers. I have to proove that this function defines a norm $R^2$. First of all, I said that, by linearity : $$N(\...
1
vote
1answer
28 views

Prove convexity of the given function on $\mathbb{R}^n$

$f:\mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$ is given by $$f(x)=\mbox{max}\{ (z_1-x_1)^{+}, (z_2-x_2)^{+}, ..., (z_3-x_3)^{+} \}$$ where $x=(x_2,x_2,...,x_n) \in \mathbb{R}^n$ and ...
1
vote
1answer
44 views

“Norm of norms” is another norm?

Suppose that, for some finite-dimensional real vector space $\Bbb R^n$, that $n_1(v)$, $n_2(v)$, ..., $n_k(v)$ are a set of norms on the space. Given some $v$, then, we can look at the "vector of ...
0
votes
0answers
26 views

Norm $C^1$ multivariate functions

Given an open and bounded subset $\,\Omega\subset\mathbb{R}^n\,$ and the interval $\,[0,T]\,$ such that $\,T<\infty$, I'm trying to find a norm for $\,X=\mathcal{C}^1\left([0,T]\times\Omega\right)$ ...
1
vote
1answer
25 views

For A,B subsets of a Normed Vector Space, A closed, B Compact, Show A - B Is Closed [duplicate]

Statement of the problem: Let $E$ be a Normed Vector Space over the real numbers. Let $A, B$ be subsets of $E$ such that: $A$ and $B$ are non-empty, $A \cap B = \emptyset $. Assume $A$ is closed and ...
1
vote
1answer
22 views

The relationship among different types of fundamental spaces.

I'm just looking to make sure my understanding of certain fundamental spaces are correct. Denote the set of all vector spaces by $V$, the set of all metric spaces by $M$, the set of all normed spaces ...
0
votes
0answers
26 views

How to prove Neumann series doesnt converge when spectral radius > 1?

For an operator $T \in B(X)$, its spectral radius is $$r(T) = \lim_{n\rightarrow \infty} \|T^n\|^{1/n}$$ and the Neumann series is $$\sum_{n=0}^{\infty}T^n.$$ If $r(T)>1$, I can show that series ...
1
vote
1answer
36 views

A necessary and sufficient condition on a normed space $X$ for the strong operator topology and norm topology on $B(X)$ to coincide

Is there a simple criterion that determines whether the strong operator topology and norm topology on $B(X)$ coincide, when $X$ is a normed space? If $X$ is a Hilbert space, the necessary and ...
4
votes
1answer
54 views

Uniqueness of Hahn-Banach extensions for $c_0 \subseteq \ell^\infty$

Let $c_0$ be the space of real sequences converging to zero with supremum norm. $c_0$ is a (closed) subspace of $\ell^\infty$, the space of bounded real sequences. A $f \in {c_0}^*$ corresponds to a $...
0
votes
0answers
42 views

Convexity properties of a cone

Let $M$ be a normed space and $A$ a subset (nonempty) of the unit sphere ($S_1$, which is the points of norm $1$). Define, for $\alpha >0$, a $A^{\alpha}=\{x\in S_1 | d\left( x,A\right) \leq\alpha\}...
0
votes
0answers
14 views

Are normed spaces equipped with the weak topology sequential [duplicate]

Let $X$ be a normed space equipped with the weak topology. Is $X$ a sequential space (using this definition)? That is can we test closedness in the weak topology using weakly convergent sequences? I ...
1
vote
0answers
100 views

When $|a-b|=|a|-|b|$?

Could someonoe help me to decide if the following satetement is true? If $K$ is a strictly convex Banach space and $a,b\in{K}$ verify $|a-b|=|a|-|b|=1$ then, $a=\lambda{b}$ for some $\lambda\geq{0}$. ...
0
votes
1answer
33 views

Closure of c$_0$ in Uniform Topology

The Uniform Topology is generated by the metric $d$ on the set $X=l^\infty$ $d(x,y)= \sup \{ |x_n -y_n|: n\in \mathbb{N}\} $ We now prove $\bar{ c_0} \subset c_0 $, let $x \in \bar{c_0}$ we have ...
0
votes
1answer
18 views

Finite Rank Operator in Normed Space, not necessarily Hilbert neither Banach

Suppose that $E$ and $F$ are normed spaces and $T:E \rightarrow F$ is a bounded linear operator. I NEED TO SHOW WHAT FOLLOWS: If there are $n\in \mathbb{N}, f_{1}, ..., f_{n}\in E^{\ast}$ (dual of $...
0
votes
1answer
30 views

Proving that $l_\infty$ is complete

I'm learning about Hilbert spaces and operators theory, from some book. I came across the following question - And the books' answer: What I don't understand in the proof - Why can we understand ...
2
votes
1answer
39 views

Banach space inequality

I'm looking to prove the following inequality \begin{align} ||\frac{u}{||u||}-\frac{v}{||v||}|| \leq 2||u-v|| \end{align} where $u$ and $v$ are elements of a Banach space such that $||u||$ and $||v|...
0
votes
0answers
4 views

Simplifying induced matrix norm expressions

Notation Absolute value of $c\in\mathbb{C} = \left|c\right|$. Entrywise absolute value of $A = \text{abs}(A)$. Complex conjugate transpose of $A = A^* = (\overline{A})^T$. $A$'s $i$th singular value $...
0
votes
2answers
33 views

The rotation matrix is non-expansive

Define for each $\alpha\in \mathbb{R},$ $$A(\alpha)=\begin{bmatrix}\cos\alpha& -\sin\alpha\\ \sin\alpha &\cos\alpha\end{bmatrix}.$$ Then $A$ is non-expansive if $$\|A(\alpha) -A(\beta)\|\leq \|...
0
votes
0answers
48 views

Are the spaces $T_pM$ and $\mathbb R^n$ homeomorphic?

Let $T_pM$ be the tangent space at a point $p$ in a n-dimensional smooth manifold $M$. In addition, if we assume $(M,g)$ as a smooth Riemannian manifold, then $T_pM$ is a n-dimensional real normed-...
0
votes
1answer
36 views

Why are reflexive spaces called like that?

A normed spaces $(X, \| \cdot \|)$ is called reflexive if the evaluation map $X \to X^{**}$ is an isomorphism. If $X$ is reflexive, it's not analytically distinguishable from it's bidual space $X^{**}$...
2
votes
1answer
49 views

The strong, weak and weak-star topologies coincide on finite dimensional spaces.

Let $E$ be a finite dimensional normed linear space. I have been able to show using set inclusion that $s=w=w^*$, where $s,\,w$ and $w^*$ represent strong, weak and weak-star topologies, repsectively. ...
2
votes
2answers
99 views

Differentiability, linear operators

Let $Y$ be a complete normed linear space, and let $M$ denote the space of bounded linear operators from $Y$ to itself. Let $L : M → M$ be the map defined by $L(A) := A^2$. I am supposed to show that ...
0
votes
0answers
32 views

Alternatives to Gram-Schmidt

So I'm curious if what I'm about to say is well-known and/or true, because I don't really have time to investigate right now. By inspecting the graphs of sets of orthonormal polynomials (like ...
0
votes
0answers
39 views

Proof of the Alaoglu Theorem

I was reading through the proof of the Alaoglu theorem which states Let $X$ be a normed space Then the unit ball in $X^*=B^*$ is compact with respect to the $weak^*$ topology. The proof goes as ...