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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Direct sum of vector space

How can give me two or three example about direct sum of vector space But I want the vector is equipped with an inner product
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35 views

For a normed vector space, prove, without the Hahn-Banach Theorem, that if $ L(x) = 0 $ for every continuous linear functional $ L $, then $ x = 0 $.

Let $E$ be a normed space. Is it true that if $L(x_0) = 0$ for every $L \in E'$ then $x_0=0$? One way to prove it is to consider $L \in E'$ s.t. $\|L\|= 1 $ and $L(x_0)= \|x_0\|$ (Existence of such ...
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2answers
14 views

Some question about orthogonal complement

Let $H$ be a Hilbert space and $Y$ is a closed subspace we denote $Z$ for a orthogonal complement of $Y$ How can I prove that $Z$ is a closed subspace of $X$ (I want to prove subspace and closed ) ...
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1answer
31 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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12 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
22 views
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12 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? ...
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1answer
23 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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3answers
41 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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1answer
10 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?
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92 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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189 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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1answer
36 views

Prob. 2, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $M$ be the subset of $\mathbb{C}^n$ such that $M$ consists of all $n$-tuples of $y = (\eta_1, \ldots, \eta_n)$ of complex numbers such that $\sum_{i=1}^n \eta_i = 1$. Then we can show that $M$ is ...
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1answer
30 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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2answers
23 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
16 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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1answer
16 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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28 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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0answers
22 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
35 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 ...
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53 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
36 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}} ...
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634 views

About an extension of Riesz' Lemma for normed spaces

The Riesz' Lemma is as follows: Let $Y$ and $Z$ be subspaces of a normed space $X$ of any dimension (finite or infinite) such that $Y$ is closed (in $X$) and is also a proper subset of $Z$. Then for ...
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2answers
100 views

Lemma 2.4-1 in Erwin Kreyszig's “Introductory Functional Analysis with Applications”: Is there an easier proof?

Here's the statement: Let $\{x_1, \ldots, x_n \}$ be a linearly independent set of vectors in a normed space $X$ (of any dimension). Then there is a real number $c > 0$ such that for every choice ...
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1answer
60 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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1answer
42 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 ...
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2answers
326 views

Finite dimensional subspace of $C([0,1])$

Let linear $S$ be a subspace of $C([0,1])$, i.e., the continuous real-valued functions on $[0,1]$. Assume that there exists $c>0$, such that $\|\,f\|_\infty\leq c \|\,f\|_2$, for all $f\in ...
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1answer
46 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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42 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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2answers
57 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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1answer
317 views

Convergence in normed spaces

I consider a sequence of elements $\{f_n\}_n$ in a normed space $X$ such as $\Vert f_n-f\Vert_X\to 0, n\to\infty$. Let $\{Q_n\}_n$ a complex sequence with $Q_n\to Q$ as $n\to\infty$. Does $Q_nf_n$ ...
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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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1answer
22 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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1answer
145 views

Operator norm of a convolution

Consider the operator on $L^2(\Bbb R)$, $f\rightarrow f*g$, where $g\geq 0$ is some $L^1$ function. Show the operator is a bounded linear operator with operator norm equal to $||g||_1$. Showing ...
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2answers
34 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 ...
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0answers
15 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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3answers
54 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
29 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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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
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
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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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 ...
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0answers
41 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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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
34 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 ...
3
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1answer
234 views

convergence of sequence of averages the other way arround

In a vector normed space, if $ \{x_n\} \longrightarrow x $ then $ z_n = \dfrac{x_1 + \cdots+x_n}{n} \longrightarrow x $ Is it true the other way arround too? meaning: if $ z_n = \dfrac{x_1 + ...
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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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4k views

Why are $l^{\infty}$ and $L^{\infty}$ non separable spaces?

$l^p (1≤p<{\infty})$ and $L^p (1≤p<∞)$ are separable spaces. What on earth has changed when the value of p turns from a finite number to ${\infty}$? Our teacher gave us some hints that there ...