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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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
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1answer
99 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 \| = \| ...
2
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1answer
26 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. ...
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2answers
85 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
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1answer
43 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 ...
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1answer
30 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.
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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
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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 ...
0
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1answer
24 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
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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 ...
1
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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 ...
0
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1answer
35 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
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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
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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
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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
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1answer
23 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 ...
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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 ...
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 ...
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1answer
83 views

Linear and nonlinear operator on normed space and its properties

My first question is : We know every bounded linear operator is continuous operator , and every continuous linear operator is bounded operator , in other words the continuousness and boundedness are ...
1
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2answers
34 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
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1answer
11 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 ...
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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 ...
0
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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 ...
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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 ...
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0answers
88 views

Properties maintained by the direct sum of normed spaces.

Let $(X_i)_{i=1}^\infty$ be a sequence of normed spaces. We define the $\ell_p$-direct sum $[\bigoplus_{i=1}^n X_i]_p$ as the normed space of elements $(x_i)_{i=1}^n\in \prod_{i=1}^n X_i$ with norm $$ ...
3
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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
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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 ...
2
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1answer
48 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 ...
1
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1answer
53 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 ...
4
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1answer
2k views

About Banach Spaces And Absolute Convergence Of Series

How to prove the following two assertions: If in a normed space $X$, absolute convergence of any series always implies convergence of that series, then $X$ is a Banach space. In a Banach space, ...
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 ...
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2answers
57 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
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2answers
226 views

Norm equivalence (Frobenius and infinity)

I am trying to prove the matrix norm equivalence for norms 1, 2, $\infty$ and Frobenius. I have managed to solve find the constants for $||.||_{1}$ and $||.||_{2}$ but I cannot see how to continue ...
2
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0answers
74 views

How to show that the normed space $(\ell^2, \|\cdot\|_2)$ is complete

Show that the normed space $(\ell^2, \|\cdot\|_2)$ is complete. I am not sure where to even start with this question. I'm quite sure it involves Cauchy and also Bolzano-Weierstrass but I'm not sure ...
3
votes
1answer
427 views

Equality in Young's inequality for convolution

I am trying to understand how sharp Young's inequality for convolution is. The inequality says $||f \ast g||_r \leq ||f||_p ||g||_q$ where as $1/p+1/q = 1+1/r$. Actually, there are a couple of papers ...
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
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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 - ...
2
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1answer
57 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
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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 ...
3
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1answer
31 views

Proving that a subspace of $L^2$ is closed.

Suppose $Z$ is a random variable on a probability space $(\Omega, F, P)$. $M(Z)$ is the subspace of $L^2$ consisting of all random variables in $L^2$ which can be written in the form $\phi(Z)$ for ...
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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 $\rho(x,y) \leq \rho(x,z) + \rho(y,z)$ ...
5
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4answers
233 views

Why are we defining the norms on certain vector spaces the way they are?

What's the intuition behind defining $\|x\|_{\infty} = \max_{1 \le i \le n}\{|x_i|\}$ on the space of ordered $n$-tuples of complex numbers? I'm asking because I've been asked to find a norm on the ...
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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
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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 ...
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0answers
14 views

Spaces where distance between a point and a hyperplane is always reached

In this article I give an example of a Banach space where the distance between a point and a hyperplane is not reached. In the same article, I mention that for finite dimensional vector spaces or ...
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1answer
31 views
1
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0answers
13 views

condition to be a norm on a finite dimensional complex vector space

Here x is an element of C^n, where n is a natural number and C is the complex number field, and p is any positive number. This becomes a norm when p>=1. But, what happens when p is between 0 and 1? ...
0
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1answer
31 views

Banach algebra norms on $M_n(A)$

Let $A$ be a Banach algebra (not sure whether I need $A$ to be unital). I saw the claim that all Banach algebra norms on $M_n(A)$ with continuous projections on entries are equivalent. How does one ...
1
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3answers
45 views

Injectivity of $T:C[0,1]\rightarrow C[0,1]$ where $T(x)(t):=\int_0^t x(s)ds $

Prob. 2.7-9 in Erwin Kreyszig's "Introductory Functional Analysis with Applications": Is this map injective? Let $C[0,1]$ denote the normed space of all (real or complex-valued) functions defined and ...
0
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1answer
11 views

Is the completion of a separable normed linear space is also separable?

Let $X$ be a separable normed linear space. Is the completion of $X$ is a separable Banach space?