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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16 views

Prove that $min\{\|x-y\||y\in M\}=max\{|\langle x,y\rangle||y\in M^\perp , \|y\|=1\}$

Suppose $M$ is a closed subspace of a Hilbert space $X$. Let $x\in X$. Prove that $min\{\|x-y\||y\in M\}=max\{|\langle x,y\rangle||y\in M^\perp , \|y\|=1\}$ My Try: First of all I am confused ...
3
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1answer
33 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}} ...
4
votes
1answer
50 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 ...
3
votes
1answer
33 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 ...
1
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1answer
41 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 ...
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2answers
52 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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3answers
36 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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0answers
25 views

How can I find the closure of $P[a,b]$ [closed]

Let $P[a,b]$ the space of all polynomials on the interval $[a,b]$ clearly $P[a,b]$ is a subspace of $C[a,b]$ but how can find the closure of $P[a,b]$ , In special case $[0,1]$ .
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1answer
19 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: ...
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2answers
39 views

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: ...
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1answer
47 views
0
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0answers
14 views

Linear functional on bounded functions in $\mathbb R$ that is translation invariant

This should be a simple problem but I can't figure it out. I'm attempting to construct a finitely additive measure on $\mathbb R$ which equals the Lebesgue measure on Lebesgue measurable sets, ...
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2answers
33 views

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 ...
2
votes
3answers
51 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 : ...
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2answers
37 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
28 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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0answers
13 views

Weak-* closure and convexity

I'm trying to write a proof of Goldstine's theorem : the weak-* closure of the unit ball of a normed vector space $X$ is the unit ball of the second dual $X^{**}$. At some point I would like to use ...
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1answer
68 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?
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0answers
23 views

Is the countable product of normed spaces a normed space with this “norm”

Consider the product $\mathcal{N} = \mathcal{N}_{1} \times \mathcal{N}_{2} \times \mathcal{N}_{3} \times \dots$ where $\mathcal{N}_{1}, \mathcal{N}_{2}, \mathcal{N}_{3}, \dots$ are normed spaces with ...
0
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0answers
36 views

Showing that if any non-zero $f \in X^*$ takes its maximum value on the unit sphere at most once, then X is strictly convex

Let $(X, \| \|)$ be a normed space. I'm trying to show that if all non-zero $f \in X^*$ take their maximum value on the unit sphere at most once, i.e. $\forall f \in X^* - \{0\}$ there is at most one ...
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0answers
29 views

$d(x,L)=\max\{f(x) \,| \, f\in L^{\perp},\, \|f\|=1\}$

Let $X$ be a normed space and $L$ its subspace. Let $L^{\perp}$ be a set of all functional of whose kernel contains $L$. Then $d(x_0,L)=\max\{f(x_0) \,| \, f\in L^{\perp},\, \|f\|=1\}$ I read a ...
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1answer
24 views

A question about the natural identification between $Y$ and $Y^{**}$, with $Y$ normed space. Is the following fact obvious?

Let $X$ be a reflexive normed space, $Y$ a normed space and $T: X \to Y$ a linear operator. I consider a sequence $\{x_n\}$ in X and the sequence $\{T(x_n)\}$ in $Y$. I know that there is a ...
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0answers
26 views

the second derivative for a composite function

Let $f: U\rightarrow \mathbb{R}$ where $U\subseteq X$ a normed vector space and suppose $f(\mathbf{u})\in C^{2}$.Let $\mathbf{x}\in U$ and let $r>0$ be such that $B(\mathbf{x},r)\subseteq U. $ ...
2
votes
1answer
61 views

Solutions to $u \circ v - v \circ u = \mathrm{Id}$

Let $(E,\Vert \cdot \Vert)$ be an infinite-dimensional normed vector space and $\mathcal{L}_{c}(E)$ denote the ring of continuous endomorphisms of $E$. I would like to determine whether the equation ...
1
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1answer
46 views

Linear and nonlinear operator on normed space and its properties

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 equivalent in linear ...
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1answer
22 views

Prob. 3, Sec. 2.8 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $C[-1,1]$ denote the normed space of all (real or complex-valued) functions defined and continuous on the closed interval $[-1,1]$ on the real line, with the norm given by $$\Vert x ...
1
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1answer
27 views

$H_\infty$ norm of a transfer function by Matlab

Consider the simplest case: $$H(z) = \frac{z+1}{z+2}$$ I use two methods to find $H_\infty$-norm: ...
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0answers
37 views

Stuck because a possible error in the statement of a functional analysis exercise.

I was given this exercise: Let $X$ and $Y$ be two normed spaces, with $X$ a reflexive space. Suppose that $T:X\rightarrow Y$ is an operator such that: $\lim\limits_{n\to\infty}x_n{=}0_X$ ...
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0answers
16 views

When is a norm of identity one?

Is there a specific condition that makes a norm (any norm) of identity equal to one in any Banach spaces? Thanks.
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0answers
27 views

Is this fact true $T:X \longrightarrow Y$ linear and bounded operator with $X$ reflexive then $T$ is compact?

I wonder if this fact is true. I consider a linear operator $T:X \longrightarrow Y$, with $Y$ and $X$ two normed spaces. I suppose that $X$ is a reflexive space and that $T$ a bounded operator. Is ...
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1answer
24 views

Nonlinear discontinuous convex function

Let $X$ be a normed vector space. Construct nonlinear discontinuous convex function $f:X \rightarrow \mathbb{R}$. I tried something with $\frac{-1}{\|x\|}$ but had no success.
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1answer
17 views

Intersection of a Hyperplane and a Subspace

Let $Y$ be a dense linear subspace of a normed space $X$, and let $M$ be a closed hyperplane in $X$. I'm trying to show that $M \cap Y$ is a hyperplane in $Y$ and dense in $M$. I've been trying to ...
2
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1answer
38 views

Banach space of p-Lipschitz functions

Given $p\in\mathbb{R}$, consider the space: $$ Lip(p) = \left\{f:[0,1] \longrightarrow \mathbb{R} : \mbox{ $f$ is $p$-Lipschitz} \right\}$$ i.e.: there is $M>0$ such that ...
2
votes
3answers
73 views

Infinite dimensional spaces other than functional spaces

"Functional analysis" is the study of infinite dimensional spaces equipped with inner product, norm, topology...etc. The most interesting spaces are the spaces of functions/operators and sequences. I ...
14
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4answers
1k views

How to develop an intuitive feel for spaces

I'm a physicist who's currently delving deeper into what I would call more 'hardcore' maths (e.g. FEM and control theory). Every now and then, I come across various spaces, such as vector spaces, ...
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1answer
28 views

What does this assumption imply in case $X$ is a complex inner product space?

If $X$ is a real inner product space and $x,y\in X$ satisfy $\|x\|=\|y\|$, then $(x-y)\perp (x+y)$. What does this assumption imply in case $X$ is a complex inner product space? My Work: I proved ...
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1answer
31 views

How to write Holder's inequality for random vectors?

For $1 < p,q < \infty$ satisfying the constraint $1/p + 1/q =1$ and for $X, Y$ random variables such that $\mathbb E [\vert X \vert ^p ], \mathbb{E} [\vert X \vert ^q ] < \infty $ we have ...
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1answer
40 views

Why is $A:X\to Y$ linear between two normed spaces is continuous iff bounded?

Why is it that every linear operator $f: \mathbb R^n \to \mathbb R^m$ is bounded and therefore continuous, but why is it that $A:X \to Y$ between two normed spaces is continuous iff bounded? That ...
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1answer
58 views

Prob. 14, Sec. 2.7 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $A \colon = [\alpha_{ij}]_{m\times n}$ be a given $m \times n$ matrix of real numbers. Let $\mathbb{R}^n$ be the normed space of all ordered $n$-tuples of real numbers with the norm defined as ...
4
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1answer
45 views

Compute $\| T \|$ with the norm $\| \cdot \| := \max_{j=1,\ldots,n} (|\cdot_j|)$

Let $A \colon = [\alpha_{ij}]_{m\times n}$ be a given $m \times n$ matrix of real numbers. Let $\mathbb{R}^n$ be the normed space of all ordered $n$-tuples of real numbers with the norm defined as ...
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1answer
41 views

Upper bound for norm of matrix (cf. Example 2.7-7 in Erwine Kreyszig's book)

Let $A \colon = [\alpha_{ij}]_{m\times n}$ be a given $m \times n$ matrix of real numbers. Let $\mathbb{R}^n$ be the norm space of all ordered $n$-tuples of real numbers with the norm defined as ...
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1answer
45 views

Holder's inequality with expectation norm

One defines the ``p-expectation norm" of a function $f$ as $\vert f \vert_p = (\mathbb E ( f^p) )^{ \frac{1}{p}}$. What is the intuition for this norm? Now why are the following true? $\vert ...
2
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1answer
27 views

Show that dual space of $R^n$ with norm 3 is equal to the $R^n$ with norm 1.5.

How can one prove that dual space ($R^n$,$||.||_3$)*= ($R^n$,||.||1.5). How to go about using the holder's inequality? Any help will be appreciated! Hint: I know I've to use holder inequality to make ...
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0answers
28 views

What is the norm of the pre-multiplication by a fixed matrix operator?

Let $A \colon= \left(\alpha_{ij} \right)_{m\times n}$ be a given $m \times n$ matrix of complex numbers, and let the operator $T \colon \mathbb{C}^n \to \mathbb{C}^m$ be defined by $$T(x) \colon= Ax ...
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0answers
19 views

Let $V$ be a NLS (over $\mathbb R$ ) of dimension $>1$, then for any $0 \le a < b$ , is the set $\{v \in V : a < ||v|| < b\}$ connected?

Let $V$ be a normed linear space (over $\mathbb R$ ) , then for any $0 \le a < b$ , is the set $\{v \in V : a < ||v|| < b\}$ connected ? I know that if $V$ is the space of complex numbers ...
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1answer
20 views

show that the normed linear vector space $\mathbb{R}^n$ with norm $||x|| = \max{(|x_1|, |x_2|, … , |x_n|})$ is complete.

I am trying to show that the normed linear vector space $\mathbb{R}^n$ with norm $||x|| = \max{(|x_1|, |x_2|, ... , |x_n|})$ is complete. My approach was as follows: First, construct a Cauchy ...
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1answer
29 views

Show that a linear form $\mathbb{R}^n \to\mathbb{R}$ is continuous

$f(x)$=$n∑k=1$ $g$($x_k$) ou $x_k$ is the kth component of the vector x. $x_k=\langle e_k,x\rangle$. I have the option of showing this with sequences (which I dont know how, I never understood how to ...
2
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2answers
49 views

$T$ linear operator s.t. $\lim\limits_{n\to\infty}x_n{=}0_X$ $\Longrightarrow$ $\lim\limits_{n\to\infty}T(x_n){=}0_Y$ then $T$ is bounded

Let $X$ and $Y$ be two normed spaces, with $X$ a reflexive space. I suppone that $T:X\rightarrow Y$ is an operator such that: $\lim\limits_{n\to\infty}x_n{=}0_X$ $\Longrightarrow$ ...
1
vote
1answer
17 views

Prove that $S:F \to \mathcal{L}(E,F)$ is a topological isomorphism

Let $E$ and $F$ be normed spaces, $E \neq \{ 0 \}$ and $x_0 \in E \backslash \{ 0 \}$, $x_0 \in E'$ such that $x_0'(x_0)=1$. Prove that the function $S: F \to \mathcal{L}(E,F)$, $S(y)=T_y$ defined ...
1
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1answer
33 views

How can I fix this proof using transfinite induction of the existence of bases of normed vector spaces?

I want to prove that every normed vector space has a basis. The following proof relies on the principle of transfinite induction. I believe that it is flawed because I'm not so sure if it's possible ...