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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2
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0answers
26 views

Connected set in normed space

I have this exercise: "let $E$ be a normed space and $X\subset E$ $$X~\text{connected}~\Longleftrightarrow \forall A\subset X,~\text{such that} A\neq\emptyset, A\neq X~\text{we have}~ Fr(A)\neq ...
1
vote
0answers
34 views

Theorem 3.6-2 in Erwine Kreyszig's “Introductory Functional Analysis with Applications:” Does the converse hold if the space is not complete? [duplicate]

First, a definition: Let $X$ be a normed space. A subset $M (\neq \emptyset) \subset X$ is said to be total in $X$ if the span of $M$ is dense in $X$. Now theorem 3.6-2 in Kreyszig states the ...
0
votes
1answer
31 views

Can't work out if this proof is sound or not. Any ideas?

Let $V$ be a normed space over some field $\mathbb K$. I proved that $$ \overline{B_r(a)} = \{v \in V \mid \|v-a\| \le r \}$$ $\subseteq $ was easy but for the $\supseteq$ direction I am really not ...
0
votes
0answers
19 views

Proof of dual norm relation: $\frac{1}{q} + \frac{1}{r} = 1$

Recall: $\|\|$ is a norm in $R^n$, and its dual norm is defined as $\|z\|^*=\text{sup}_{\|x\|\leq1}z^Tx$. If $q$-norm and $r$-norm are dual norm, then we have the following relation: ...
0
votes
0answers
12 views

Mean Value Inequalities for vector-valued functions

Let $X$ and $Y$ be Banach spaces, and let $U\subset X$ be open. If $f\colon U\to X$ is continuously differentiable and $x,v\in X$ are such that the line segment $\ell=\{x+tv\mid t\in[0,1]\}$ lies ...
1
vote
2answers
28 views

Determine interior and boundary of a set

Let $(X,||\cdot||)$ be a normed vector space, where $$X = \big\{ (a_n)_{n \geq 1} ~~|~~ (a_n)_{n \geq 1} \text{ is a bounded real sequence }\big\}$$ and $$\|(a_n)_n\| = \sup_{n \in ...
1
vote
0answers
20 views

finding Interior Points of a set

In the normed space $(\mathbb{R}^2, ||(x_1,x_2)||:=|x_1|+|x_2|)$ I want to find Interior Points of $$ \{ (x,1/n) ~~\big|~~ x\in \mathbb{R} \text{ and } n\in \mathbb{N} \}. $$ I guess that the ...
0
votes
1answer
44 views

The set of infinite sequences with finitely many nonzero values is dense.

Could I get a proof to this lemma or a reference if a proof is too time consuming?
0
votes
0answers
27 views

What is the norm for the product of normed spaces?

Suppose $(\Omega, \Sigma ,\lambda)$ is a probability space and $X_i=L^2(\Omega,\Sigma,\lambda ,[0,1])$ with norm $||.||_{L^2}$ for all $i\in I$, $I$ is finite. Is there any natural norm for the ...
1
vote
2answers
32 views

“isomorphic” normed spaces and reflexivity

Let X, Y be normed spaces and suppose that there exists an bijective isometry between them. And if X is reflexive, then it is intuitively clear that Y is reflexive also. But, when I tried to prove ...
-1
votes
0answers
22 views

Prob. 8, Sec. 3.2 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS [closed]

if $\|x+ \alpha y\|\ge \|x\|$ for all $\alpha$, then show that $x$ and $y$ are perpendicular. If you could explain this I shall be thankful....
4
votes
1answer
24 views

Unit ball separable $\Longrightarrow$ Space separable

Given a normed space $X$ and assume it is also a locally convex space in some other topology (e.g. weak or weak* if it's a dual). Assume that the unit ball $B_X$ is separable in this topology. Is it ...
0
votes
1answer
26 views

Two-dimensional subespace suplementary of another one

Let $E$ be a real normed space. All subspaces $S$ of codimension $1$ (hyperplane) in $E$ is either, closed or dense. What do say about a similar property when $S$ is of codimension $2$?
2
votes
1answer
30 views

Normed space where unit ball's weak and norm topology coincide?

Let $X$ be a normed space. Are there any infinite-dimensional examples such that the $(B_X, w\restriction_B) = (B_X, \|\cdot \|\restriction_B)$. In finite dimensions this obviously holds always true. ...
4
votes
1answer
120 views

Can the sum of two measurable functions be non-measurable if they are valued in a general normed space instead of $ \mathbb{R} $?

It's well known that the sum of measurable functions is measurable, if they are real or complex valued. However, the proofs I've seen heavily rely on the usage of the countable set of rational ...
3
votes
1answer
119 views

A name for the property $ \| x \star y \| = \| x \| \| y \| $.

Suppose that $ \star: V \times V \to V $ is some binary operation on a vector space $ V $. Should it hold, is there a name for the following property? $$ \forall x,y \in V: \quad \| x \star y \| = \| ...
1
vote
1answer
61 views

Linear application on a normed space

How to prove that if $E$ and $F$ are tow normed spaces with $dim(F)<\infty$ and $f\in L(E,F)$ Then $$f~ \text{open} \Longleftrightarrow f ~\text{surjective}$$ If i suppose that $f$ is open, then ...
-1
votes
1answer
31 views

Prove norm does not come from inner product.

I know I have to show it does not satisfy the parallelogram law but I don't know how to apply it.
0
votes
1answer
33 views

Test for normability of a metric on a Banach space

If d is a metric on a (finite dimensional) Banach Space and there exist norms $\|-\|_1$ and $\|-\|_2$ and constants $C_1,C_2 \in [1,\infty)$ satisfying: \begin{equation} C_1\|x-y\|_1 \leq d(x,y) \leq ...
1
vote
1answer
17 views

$x=argmin_{x\in A}||y-x||_2$ iff $\langle y-x,z-x\rangle \leq0$ for all $z\in A$

Consider $x\in A\subset\mathbb{R^n}$ with A closed and convex. How can you see that $$x=argmin_{x\in A}||y-x||_2$$ iff $$\langle y-x,z-x\rangle \leq0$$ for all $z\in A$. I tried using ...
0
votes
1answer
25 views

Estimation of the integral

I am trying to compute, or find a good estimate from above the following integral $$ \frac{1}{\pi}\int_{-\infty}^{\infty}|t|^{-1/p}\left|\frac{|t|^{\nu}-1}{t-1}\right|dt, $$where $0<1/p<1$ and ...
3
votes
5answers
49 views

Equivalence of norms: $\|x\|$ and $\|x\|_1=\|x\|+\lvert\,f(x)\rvert$

Let $f: X\to \mathbb{R}$ be a linear functional and $\|\cdot\|_1$ is defined as follow $\|x\|_1=\|x\|+|f(x)|$. Prove or disprove $\|\cdot\|_1$ is equivalent to $\|\cdot\|$ iff $f$ is continuous. I've ...
0
votes
1answer
39 views

Prob. 2, Sec. 3.3 in Erwin Kreyszig's “Introductory Functional Analysis With Applications”: How to minimise the norm?

Let $z$ be a given complex number. Let $M \subset \mathbb{C}^n$ be given by $$M \colon= \left\{ (\xi_1, \ldots, \xi_n ) \in \mathbb{C}^n \mid \sum_{i=1}^n \xi_i = z \right\}.$$ Then $M$ is convex ...
1
vote
1answer
30 views

Quotient space of infinite dimensional vector space

On an exam today I used that if $X=\mathcal{C}[a,b]$ and $Y=\{f\in X : f(a)=f(b)\}$, then the projection $\pi: X\rightarrow X/Y$ has the property $\ker(\pi)=Y$. This led me to the following: Suppose ...
1
vote
1answer
22 views

Subset of the sequence space that's closed and bounded but not compact

Consider the sequences space $l^1 = \{a = (a_n)_{n \in \mathbb{N}_0} \subset \mathbb{C}, \sum_{n = 0}^\infty|a_n|< \infty\}$ with the norm $||a||_1 = \sum_{n = 0}^\infty|a_n|$. I want to show that ...
1
vote
1answer
21 views

Show that a subset of $(\mathbb R^n,||.||)$ is closed

Let $C$ be a closed subspace of the normed linear space $(\mathbb R^n,\| \cdot \|)$.Let $r(>0)\in \mathbb R$ Define $D:=\{y:\exists x\in C$ such that $\|x-y\|=r\}$. Show that $D$ is closed. My ...
2
votes
0answers
31 views

a norm is symmetric if and only if it is unitarily invariant

how can I prove this : A norm on $\mathcal{M}_n(\mathbb{C})$ is symmetric if and only if it is unitarily invariant ? My attempt I know that a symmetric norm is a norm which verifies : $$N(ABC)\leq ...
2
votes
1answer
24 views

A question on Isometry between the orthogonal subspaces of Hilbert spaces

I was reviewing my class-notes on Functional analysis and the professor had mentioned that given a closed proper subspace $U$ of an hilbert space $\mathcal{H}$, $\exists $ a closed subspace ...
-2
votes
1answer
67 views

Let we have the following exercise [closed]

How can I solve the following exercise Let $T:C[0,1]\rightarrow C[0,1]$ be defined by $$Tx(t)=y(t)=\int_0^t x(\tau)d\tau.$$ Find $\mathscr R(T)$ and $T^{-1}:\mathscr R(T)\rightarrow C[0,1]$. Is ...
3
votes
2answers
37 views

What is the range of the operator $T$ I mean I want to determine $R(T)$

Given the normed space $\ell^\infty$ of all bounded sequences of (real or complex) numbers with the norm given by $$||x||:= \sup_{j\in Z^+} |\xi_j|,$$ for each $x:=(\xi_j)_{j=1}^\infty$ in ...
-2
votes
2answers
87 views

I want some help in functional analysis [closed]

I want sone help in functional analysis : $1)$ consider the vector space $X$ of all real -valued functions which are defined on $R$ and have derivatives of all orders everywhere on $R$ define ...
1
vote
2answers
35 views

Continuity of $(a_1,\ldots,a_n) \mapsto \sum_1^n a_j e_j$

Let $e_1,\ldots,e_j$ be a basis for a finite dimensional normed vector space $X$. I wish to show that the map $(a_1,\ldots,a_n) \mapsto \sum_1^n a_j e_j$ is continuous, where $(a_1,\ldots,a_n)$ has ...
1
vote
1answer
12 views

A linear map, continuous at zero, is bounded

I am studying a theorem about linear maps on normed vector spaces. The theorem states the equivalence of the following statements: $T$ is a linear map on a normed vector space $X$ that maps to a ...
1
vote
0answers
15 views

The closure of a subspace of a normed vector space is a subspace

This is a self-study problem (Folland Real Analysis exercise 5.5). If $\mathcal{X}$ is a normed vector space, the closure of any subspace of $\mathcal{X}$ is a subspace. My attempt: It is ...
2
votes
2answers
42 views

vector space of continuously differentiable functions is complete regarding a specific norm

Consider $C^1[a, b] = \{f: [a, b] \to \mathbb{C}\mid f\text{ continuously differentiable}\}$. Consider the following norm: $$\|f\|_{C^1} = \|f\|_\infty + \|f'\|_\infty$$ Now, it needs to be shown ...
0
votes
2answers
29 views

Proving a norm on the space of differentiable functions

I consider the space $C^1[a, b]$ of (complex) functions that are at least once differentiable on $[a, b]$. I want to show that $$||f||_{C^1} := ||f||_\infty + ||f'||_\infty$$ defines a norm on ...
-2
votes
1answer
79 views

Some question about linear operator on normed space [closed]

$1)$ let $X$ and $Y$ be normed space , show that a linear operator $T:X\rightarrow Y$ is bounded if and only if $T$ maps bounded sets in $X$ into bounded sets in $Y$ $2)$show that the operator ...
3
votes
2answers
54 views

Example 5, Sec. 24 in Munkres' TOPOLOGY, 2nd ed: Is this map always continuous?

Let $(X, \Vert \cdot \Vert)$ be a given normed space that has elements other than the zero vector $\theta_X$. And let $T \colon X-\{\theta_X \} \to X$ be defined by $$T(x) \colon= \frac{1}{\Vert x ...
0
votes
0answers
11 views

Are two operator norms on $M_n(A)$ equivalent?

If $A$ is a Banach algebra, then $M_n(A)$ can be given the operator norm as operators on $A\oplus_p\cdots\oplus_p A$ ($1<p<\infty$) to make it a Banach algebra. If in addition $A$ is an operator ...
-1
votes
2answers
59 views

Does this hold for $p=\infty $, i.e., is it true that $(l^{\infty})'= l^1? $ [closed]

Let $E=l^p$ where $1 \le p < \infty $ we know $E'=l^q$ Where $q$ is the dual exponent of $p$, i.e. $q$ is such that $\frac{1}{p}+\frac{1}{q}=1$ Does this hold for $p=\infty $, i.e., is it true ...
0
votes
3answers
41 views

Closed subset of the $\mathbb{R}^n$

I want to show that $U = \{(x, y) \in \mathbb{R}^2|xy ≤ 1\}$ is a closed subset of $\mathbb{R}^2$. Yes there are (easy) ways to do this using functions, but what's the (easiest) way to prove this ...
1
vote
1answer
54 views

$(B_X,w)$ metrizable implies $X^\ast$ separable

Let X be a normed space and assume that $(B_X,w)$ is metrizable, i.e. the weak topology is metrizable. Show that $X^\ast$ is separable. My attempt: Let $d$ a equivalent metric on $B_X$. For fixed ...
2
votes
1answer
49 views

a question about proving a normed space is complete

Let [a,b] be an interval in R,and denote by E the Vector space of functions f:[a,b]->R such that f is of bounded variation over [a,b] and f(a)=0.Prove that by setting $||f||=Var|_{a}^{b}(f)$ for each ...
0
votes
1answer
29 views

Is the norm on $\ell^\infty$ induced by an inner product?

Let $\ell^\infty$ be the normed space of all bounded sequences $x \colon= (\xi_n)$ of all bounded sequences of complex numbers, with the norm defined by $$\Vert x \Vert_\infty \colon= \sup_{n \in ...
0
votes
2answers
25 views

metric on the set of complex sequences

Let X be the set of complex sequences $(a_n)_{n\in\mathbb{N}}\in \mathbb{C}$. Show that the transformation: $$ d((a_n), (b_n)) := \sum_{n=0}^\infty \frac{1}{2^{n+1}} \frac{|a_n - b_n|}{1 + |a_n - ...
0
votes
1answer
22 views

equations for an open ball in a normed space

Let $(V, \|\cdot\|)$ be a normed space. Show that for an open ball $B_1(0) \subseteq V$, it holds true that: $∂B_1(0) = \{x \in V: d(x, 0) = 1\}$ where $d(x, 0) = \|x\|$. Also, figure out the ...
2
votes
1answer
58 views

Completeness of $C^1$ functions vanishing at infinity with sup-norm of derivatives

I'm looking at $$C_0^1(\mathbb{R}) := \{f \in C^1(\mathbb{R}) : \lim_\limits{|x|\rightarrow \infty}f(x) = \lim\limits_{|x|\rightarrow \infty} f'(x) = 0\},$$ along with the norm given by $||f|| := ...
0
votes
1answer
13 views

Proving distance inequality between three elements in a normed linear space

For any two elements $x,y$ belonging to a normed linear space, distance between x and y is given by $\rho(x,y) = ||x-y||$ I am trying to prove the inequality $\rho(x,y) \leq \rho(x,z) + \rho(y,z)$ ...
1
vote
2answers
61 views

For a normed vector space $ E $ and an element $ x \in E $, prove that if $ L(x) = 0 $ for every continuous linear functional $ L $, then $ x = 0 $.

Question. Let $ E $ be a normed vector space. Is it true that for a given $ x \in E $, if $ L(x) = 0 $ for every $ L \in E' $, then $ x = 0_{E} $? One way to prove this is to find an $ L \in E' ...
2
votes
1answer
36 views

How do you formally construct the following proof regarding completeness and vector spaces?

Let $S$ denote the vector space of all finitely nonzero sequences; that is, $X =(X_n) \in S$ if $X_n = 0$ for all but finitely many n. Show that $S$ is not complete under the sup norm $\| X \|_\infty ...