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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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 ...
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1answer
23 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 ...
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45 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
42 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 $U^{\...
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67 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 $\ell^\...
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49 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 ...
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1answer
31 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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38 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 ...
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309 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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48 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 $C^1[...
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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 \...
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86 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 ...
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3answers
50 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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1answer
69 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 $N\...
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1answer
76 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 ...
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1answer
48 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 \...
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2answers
30 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 - b_n|...
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1answer
29 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 ...
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1answer
176 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|| := \...
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19 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 $\rho(x,y) \leq \rho(x,z) + \rho(y,z)$ I'...
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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' $ ...
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1answer
52 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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46 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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0answers
19 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? I ...
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1answer
43 views

Is there an incomplete normed space which is Asplund?

Can there exist an incomplete normed space which is Asplund?
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1answer
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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?
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45 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 ...
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360 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
31 views

Norm on the space of sequences

Given the sequence spaces $\ell^p$ that are defined as: $$\ell^p = \left\{a = (a_n)_{n\in\mathbb{N}}, \sum_{n=0}^\infty |a_n|^p < \infty\right\}$$ for $\infty > p ≥ 1$, I'm trying to show that ...
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1answer
80 views

Equivalent definition of bounded set in norm linear space

Definition of Bounded set in norm linear space. If $X$ is norm linear space and $B \subseteq X$, then $B$ is bounded if there exists $M>0$ such that $\sup_{n\in \mathbb N} \lvert \lvert x_n \...
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1answer
360 views

Proving norm on a vector space

Let C[0,1] be the set of all continuous functions f: [0,1] -> R, Prove that ||f|| = max |f(x)| ,x in [0,1], is a norm of this vector space. In a previous exercise, I already proved that C[0,1] was a ...
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1answer
27 views

sequence spaces as subsets of each other

Given the sequence spaces $\ell^p$ that are defined as: $$\ell^p = \left\{a = (a_n)_{n\in\mathbb{N}}, \sum_{n=0}^\infty |a_n|^p < \infty\right\}$$ for $\infty > p ≥ 1$, how can it be shown ...
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124 views

Prob. 9, Sec. 3.2 in Erwine Kreyszig's “Introductory Functional Analysis with Applications”

Here is Prob. 9 in the Problems after Sec. 3.2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $V$ be the vector space of all continuous complex-valued functions on $...
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1answer
116 views

Is the norm operator between normed spaces ever induced from an inner product?

Assume $(V,\| \|_V),(W,\| \|_W)$ are both finite dimensional normed spaces. We have the induced operator norm on $Hom(V,W)$. When does it occur that this norm is actually induced from some inner ...
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1answer
105 views

$L_\infty[0,1]$ completness and separability

Prove that $L_\infty[0,1]$ with the norm given by $$\Vert f\Vert_\infty:= \inf\{S(N): \mu(N)=0\}, \quad \mbox{where} \quad S(N)=\sup\{|f(x)|: x\notin N\}.$$ is complete and is not separable....
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202 views

Continuity of metric function

Let $X$ is a vector space and $d$ is a metric function on $X$ and $\|\cdot\|$ is a norm on $X$ and $\langle\cdot,\cdot\rangle$ is an inner product function on $X$ It is to easy to prove $\|\cdot\|$ ...
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1answer
59 views

Convergence in Fréchet spaces and the topology

actually i have two questions concerning Fréchet spaces (i am not familiar with these spaces so i need some help). I have a vector space $V$ with a distance $$d(x,y) = \sum_{n\in \mathbb{N}} \frac{|x-...
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108 views

Categorical Banach space theory

Consider the category $\mathsf{NormVect}_1$ of normed vector spaces with short linear maps$^{\dagger}$ and the full subcategory $\mathsf{Ban}_1$ of Banach spaces with short linear maps. Both ...
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106 views

How can I prove that $f$ is inner product function

We know the polarization identity in inner product space : $$\langle x,y\rangle= \frac{1}{4} (\|x+y\|^2-\|x-y\|^2) + \frac{i}{4} (\|x+iy\|^2-\|x-iy\|^2) $$ But the question is if we have $(X,\|\cdot\|)...
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1answer
85 views

Orthogonal in inner product space

Let $(X,<.>)$ is an inner product space prove that $x$ and $y$ are orthogonal if and only if $||x+αy|| \ge ||x||$ for any scalar $α$ . The first direction if $x$ and $y$ are orthogonal then$||...
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149 views

Banach space it isn't Hilbert space [duplicate]

How can give me two or three examples about Banach spaces which it is not Hilbert spaces with proof ( I mean why it isn't Hilbert spaces ) ?
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70 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 ...
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37 views

The linearness of extension of linear bounded operator

Let we have the following bounded linear operator $$T: D(T)\rightarrow Y$$ such that $D(T)$ is the domain and it is a vector space and $Y$ is a Banach space . Then it has an extension $$G: W\...
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Some question about extension of bounded linear operator

Let we have the following bounded linear operator $$T: D(T)\rightarrow Y$$ such that $D(T)$ is the domain and it is a vector space and $Y$ is a Banach space . Then it has an extension $$H: E\...
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Help needed in a proof to show path connected-ness of a special kind of set in real Euclidean space

Let $A$ be a connected subset of $\mathbb R^n$ and $\epsilon >0 $ , consider $V:=V(\epsilon , A):=\{x \in \mathbb R^n : d_A(x)< \epsilon \}$ , I need to show that $V$ is path connected , so we ...
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90 views

Is every open cover of $\{(x,y) \in \mathbb R^2 : x^2+y^2=1\}$ also an open cover of some $\delta$ annulus around the unit circle?

Let $\{U_\alpha\}$ be an open cover of $\{x \in \mathbb R^n:\|x\|=1 \}$ , $n \ge 2$ , then does there exist $\delta >0$ such that $\{U_\alpha\}$ is also an open cover of $\{ x \in \mathbb R^n : 1-\...
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2answers
51 views

Continuous Linear Operator in $\mathbb{R}$- normed spaces.

Let $E$ and $F$ $\mathbb{R}$-normed spaces and let $f:E\longrightarrow F$ satisfying: $f(x+y)=f(x)+f(y) \,\forall x,y\in E$; $f$ is bounded in the unit ball $B_E = \{x\in E: \Vert x\Vert \...
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1answer
66 views

Real part of a complex function

Let $V$ be a normed linear complex vector space. If $u$ is the real part of the complex-linear functional $f$ on $V$, then $f(x) = u(x)-iu(ix)$ and $\|f\| = \|u\|$. In the proof, there is one part ...
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1answer
70 views

How do I show that this function is a contraction?

Assume $0 \lt \alpha \lt 1$. Let $f:R^N \to R^N$ with $$f_i(x) \equiv \frac{x_i^\alpha}{\sum x_j^\alpha}$$ How do I show that this function is a contraction?