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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59 views

help about supremum norm proof

T:( $C'[0,1] ,||x||₁)\to (C[0,1]$,||x||∞) Tx(t)=x'(t) for any x is in C'[0,1] $||x||₁=||x||_{\infty}+||x′||$∞ how can we prove that equality? $||x||_1 = \sum_{n=1}^{\infty}|x_{n}|$ ...
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1answer
204 views

Uniqueness of best approximation in strictly or uniformly convex normed linear spaces

I am studying approximation theory, just to learn basically, and in different books I saw a theorem about uniqueness of best approximation. One book uses strict, the other uses uniform convexity with ...
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2answers
29 views

Bounding the distance between $L_\infty$ and $L_2$ for a continuous function

Consider a set of continuous (or even differentiable) functions $f_i(x)$, all defined for $x\in [a,b]$ for $i=1\ldots,N$. Can one define a uniform constant $c$ (which may depend on $f$) such that ...
4
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3answers
116 views

Showing that a sequence converges in norm

I am trying to prove the following: Let $X$ be a normed linear space satisfying the property: $\forall \left\{x_n\right\}, \left\{y_n\right\} \subseteq X $, we have $\|x_n\|=\|y_n\|=1, ...
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1answer
62 views

Equivalence of norms given continuous identity

It is known that $\parallel \; \parallel_{1}$ & $\parallel \; \parallel_{2}$ are equivalent norms over $X$ if there are $A,B>0$ such that $A\parallel x \parallel_{1} \leq \parallel x ...
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1answer
63 views

Does completing a normed space commute with taking quotients?

Let $X$ be a normed vector space and $Y \subset X$ a closed subspace. We consider the quotient $X / Y$ and equip it with the quotient norm. Then we may form the completion $\overline{X / Y}$. We ...
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25 views

Simultaneous extension and complemented subspace

The following is Exercise 3.13.5 of Conway's Functional Analysis: Let $X$ be a compact set and let $Y$ be a closed subset of $X$. A simultaneous extension for $Y$ is a bounded linear map $T:C(Y)\to ...
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1answer
82 views

Frechet derivative of the sup norm function on $C[0,1]$

I know that the derivative in the title does not exist for any x but I do not have a clue why. Could someone explain why this derivative DNE at 0 and I can figure out why it does not exist at x? ...
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3answers
64 views

Counterexamples for Hölder's inequality when $p$ and $q$ are not conjugate.

Hölder's inequality shows that, when $$ \frac{1}{p} + \frac{1}{q} = 1,$$ and $f\in L^p$ and $g\in L^q$, then $$\Vert f\,g\Vert_1 \le \Vert f \Vert_p \Vert g \Vert_q.$$ Is there an example of this ...
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1answer
83 views

Subspaces and convergence in weak* topology

I would like to ask some questions regarding convergence in the weak* topology and subspaces. Let $X$ be a normed space with subspace $A \subset X$. Assume $X$ is endowed with the weak* topology. ...
4
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1answer
153 views

Trouble finishing a (direct) proof that $\ell^2(A)$ is a complete metric space

Let $A$ be any non-empty set. We can define summations of non-negative numbers over this index set by using a supremum of summations over finite subsets of $A$. That is, $$\sum\limits_{\alpha \in A} ...
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1answer
179 views

norm of integral operator in $C([0,1])$

If we define on $C([0,1])$ the operator $$ Tf(x) = \int_{0}^{1} K(t,s) f(s) ds$$ where $K$ is a continous function on two variables. I want to show that: $1)$ $||T|| = \displaystyle\max_{t} ...
2
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0answers
77 views

Weak sequential compactness in a reflexive space

Let $\{X, \| \cdot \|\}$ be a normed space, $B$ is the unit ball of $X$. If $\{X, \| \cdot \|\}$ is reflexive, then is $B$ weakly sequentially compact? If it's not true, are there any counterexamples ...
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1answer
150 views

$X$ complete normed space $\implies\mathrm B(X,Y)$ complete normed space?

$\newcommand{\N}{\mathbf N}\renewcommand{\leq}{\leqslant}\renewcommand{\geq}{\geqslant} \newcommand{\eps}{\varepsilon}$I was looking through the functional analysis notes of TWK (on his webpage ...
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0answers
18 views

Terminology: Contraction *of* normed spaces? *Between* normed spaces? *On* normed spaces?

I have a terminological question. Suppose $X$ and $Y$ are normed spaces, and let $f$ be a contraction $X \to Y$. Which of the following expressions is correct? $f$ is a contraction of normed spaces. ...
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0answers
51 views

Finding the norm of a linear operation.

I am reading A course in real analysis by John McDonald, on page 530, it says "it is easy to show $|||J|||=1$" where $J$ is the linear operation $J:C([0,1])\rightarrow C([0,1])$, defined by $J(f)(x) = ...
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1answer
79 views

Non-subdifferentiable convex function on a normed space

Is there any convex function $f$ on a normed space $X$ such that $\partial f(x)=\emptyset$ at every $x\in X$?
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0answers
35 views

Does this topology on the dual have a name

Let $X$ be a topological vector space. Let $X^\ast$ denote its continuous dual. It is possible to endow $X^\ast$ with the weak star topology: Def.1: If $e_x: X^\ast \to \mathbb C$ is the map $\varphi ...
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1answer
99 views

The dual of subspace of a normed space is a quotient of dual: $X' / U^\perp \cong U'$

I wanted to show that $X' / U^\perp \cong U'$, for $U$ being a closed subspace of the Banach space $X$. Therefore I looked at $l: X' / U^\perp \cong U' , x' + U^\perp=[x'] \mapsto x'|_U$. It is ...
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153 views

Strongly convex and Frechet differentiable function in reflexive Banach space

We first recall two definitions about strong convexity and Frechet differentiability in normed space. Let $(X, \|.\|)$ be a normed space and $f:X\rightarrow\mathbb{R}$ be a function. (a) $f$ is said ...
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1answer
49 views

Which of the following sets are open (or closed)?

a.) $A:= \{(x_n)_{n\in \mathbb{N}} : x_n \in [0,1] \hspace{2mm}\text{for all}\hspace{2mm} n\in\mathbb{N}\}$ in $(l^\infty, \|\cdot\|_{\infty})$ and b.) $B:= \{f\in C([0,1]) : |f(t)-t|<1 ...
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58 views

What extra assumption makes this transformation affine?

Let a vector space $V$ be given. Let $f:V\to V$ have the property that for all $x,y,a\in V$, $$ f(x+a)-f(y+a) = f(x) - f(y) \tag{$\star$} $$ Q1. I'd like to know how weak one can make additional ...
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1answer
78 views

Separability of a normed space

The following is an Exercise 12, page 75 of conway's Functional Analysis. Let $\oplus_\infty X_i = \{x\in \sqcap X_i: ||x||=\sup||x(i)||<\infty\} $ where each $X_i$ is a normed space for $i\in I$. ...
3
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1answer
133 views

Show that a normed Vector space is complete, need smart help.

I want to show that a normed vector space is complete. I know that if you can show that every Cauchy sequence converges, then it is complete. But in a normed vector space, completeness is equivavlent ...
13
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2answers
359 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 ...
0
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1answer
70 views

Homework excercise, completeness in Vector-spaces, is it correct?, long, but can it be simplified?

I have a very difficult excercise. I see now that it became too much text for someone to might go through it, if you can please help me, but don't want to read all, can you please then only answer my ...
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5answers
1k views

Why is one proof for Cauchy-Schwarz inequality easy, but directly it is hard?

Let's say you are in $\mathbb{R}^n$ and you define the norm as $||x||=\sqrt{x_1^2+x_2^2...+x_n^2}$. This we recognize as the usual norm from the inner product: $||x|| = \sqrt{\langle x, x \rangle}$, ...
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2answers
196 views

Question about Normed vector space.

Here is the definition of a normed vector space my book uses: And here is a remark I do not understand: I do not understand that a sequence can converge to a vector in one norm, and not the ...
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1answer
43 views

What norm on $\mathbb C (z)$

There are several different ways to define a norm on the space of polynomials $\mathbb C [z]$. For example, $\|p\| = \sup_{|z|\le 1}|p(z)|$ defines a norm. If $\mathbb C (z)$ denotes the field of ...
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2answers
31 views

Does one of these conditions for norms follow from the other?

The two conditions are: For all unit vectors $\mathbf{x}$ and $\mathbf{y}\hspace{-0.02 in}$, $\:$ if $\; \left|\left|\hspace{.03 in}\mathbf{x}\hspace{-0.05 in}+\hspace{-0.04 ...
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2answers
84 views

What's wrong in this reasoning of $l_\infty$ separability?

While solving a problem, related to functional analysis, I've accidentally got a "proof" of $l_\infty$ being separable, tried to find a fault in it (as the result isn't true), but didn't succeed in ...
2
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0answers
43 views

Normalizing multiple different features from unknown distributions

I'm doing some "exploratory" data analysis over a large set of classes/proteins, with a few hundred different features (I.E. Continuous variables) extracted from the data. The features are calculated ...
2
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1answer
57 views

Continuous functional that separate points

This is an exercise from Royden's Real Analysis. Let $X$ be a normed linear space and $W$ a subspace of $X^*$ that separate points. For any topological space $Z$, show that a mapping $f:Z\to X$ ...
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1answer
72 views

Show that $\lim_{p \to \infty}||x||_p=||x||_M$ [duplicate]

Let $$||x||_p=\left( \displaystyle \sum_{i=1}^n|x_i|^p\right)^{\frac{1}{p}}$$ and $$||x||_M=\max\{|x_1|,|x_2|,...,|x_n|\},$$ norms in $\mathbb{R}^n$. Show that $$\lim_{p \to \infty}||x||_p=||x||_M, \ ...
2
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1answer
130 views

Show that $L^1\subsetneq (L^\infty)^*$ [duplicate]

How does one show that $L^1\subsetneq (L^\infty)^*$? I am having trouble in this. Any help would be appreciated.
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1answer
28 views

why a lemma shows well-definedness of linear transformations

The following lemma can be used to show that some linear transformations are well-defined. I don't quite see that. I mean, if a linear transformation $T$ is well-defined, then if $x=y$ then ...
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1answer
64 views

Banach space problem

I came across the following problem: For an open set $U$ in $\mathbb{R^n}$ we define the set of all k-times continuously differentiable functions $f:U\rightarrow \mathbb{R}$ for which $D^\alpha f$ ...
0
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1answer
120 views

Covering of closed unit ball with closed balls.

Notations and definitions Let $E$ be a finite dimensional vector space with norm $||\;||$. Let $B$ denote the closed unit ball in $E$ and $B_r[a]$ the closed ball centered at $a$ with radius $r$. ...
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2answers
78 views

What does this theorem mean?

Let $(V,\|\cdot\|)$ be a finite-dimensional normed space. Define $\|T\|_\mathrm{op}=\sup\{\|T(x)\|:\|x\|≦1\}$, for all linear operators on $V$ Define $\Omega$ to be the set of all invertible linear ...
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29 views

A System of Matrix Equations

I am looking for the solutions to the following system of matrix equations: (1) $AA^{*} - A^{*}A = C^{*}C - BB^{*}$ (2) $DD^{*} - D^{*}D = B^{*}B - CC^{*}$ (3) $AC^{*} - C^{*}D = A^{*}B - ...
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62 views

Largest/smallest cross-norm: A simple question about cross-norms on tensor products of Banach spaces.

This is a very simple dumb question as I’m completely new to the topic. I was reading Wikipedia’s entry on “Topological Tensor Products”, and there’s one thing I’m confused about. Let $ A $ and $ B $ ...
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0answers
136 views

The Hilbert space $\mathcal{H}_\eta$ and unitary correspondence with $L^2[a,b]$

The question I have is related to a problem in Stein and Shakarchi's Real Analysis, Chapter 4. The problem Let $\eta(t)$ be a fixed strictly positive continuous function $[a,b]$. Define ...
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1answer
55 views

Prove scalar product in a normed vector space is an open mapping.

I have the feeling this is a really obvious question, but I'm having trouble with it, here it goes: Let $(X, ||\,||)$ be a normed vector space over $K$, prove that $\odot:K\setminus\{0\}\times X ...
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1answer
42 views

What is *the domain* of the definition of the operator norm.

My definition: Let $(V,||\cdot||),(W,||\cdot||)$ be normed spaces over $\mathbb{F}$. Let $T:V\rightarrow W$ be a continuous linear transformation. Then $||T||_{op}\triangleq ...
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2answers
43 views

How do i prove that continuous linear transformation between normed spaces is bounded?

Let $(V,||\cdot||_V),(W,||\cdot||_W)$ be normed spaces over $\mathbb{C}$. Let $T:V\rightarrow W$ be a linear transformation. Assume $T$ is continuous. Then, how do i prove that $\exists c>0$ ...
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0answers
63 views

Understanding the right assumption about Gateaux derivative in Banach spaces

It is a general fact that the notion of Gateaux derivative is not uniform over the mathematical community i.e. someone requires it to be linear and continue other not and require this additional ...
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3answers
110 views

Showing that the norm of the canonical projection $X\to X/M$ is $1$

How do I show that given $M$ a closed subspace of a normed space $X$, and let $\pi$ be the canonical projection of X onto $X/M$. Prove that $\|\pi\| = 1$. I figure I could use Riesz' lemma and set ...
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2answers
68 views

a functional analysis question

$X$ is a banach space and $f$ a non zero linear functional. I'm trying to show $null(f)$ not dense in X $\implies f$ continuous. I've tried a few approaches but I think the following seems the most ...
6
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1answer
120 views

Duality mappings on finite-dimensional spaces

I have a few questions regarding some concepts in a book "nonlinear partial differential equations by Roubicek" I am studying. The following is from the text. "Let $V$ be a separable, reflexive ...
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1answer
58 views

Are $C^0[a,b]$ and $C^0[0,1]$ isometrically isomorphic?

Consider $C^0[a,b]$ and $C^0[0,1]$, each equipped with the $L^1$-Norm. Are these (out of curiosity) isometrically isomorphic?