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.

learn more… | top users | synonyms

5
votes
6answers
414 views

Is there such thing as an unnormed vector space?

I learned about Banach spaces a few weeks ago. A Banach space is a complete normed vector space. This of course made me wonder: are there unnormed vector spaces? If there are, can anyone please ...
0
votes
0answers
29 views

Why is boundedness necessary in this proof that addition and scalar multiplication are continuous maps in the metric space induced by a norm?

Why is boundedness necessary in this proof that addition and scalar multiplication are continuous maps in the metric space induced by a norm? Suppose that $\lambda _n \to \lambda $, $\mu _n \to \mu ...
2
votes
2answers
91 views

TVS: Topology vs. Scalar Product

I just had an idea: It is clear that every scalar product induces a norm and that a metric and that finally a topology. Turning this argument around we know: Not every topology induces a metric only ...
0
votes
0answers
31 views

Commutativity and norms of specific operators (Problem 2.7.10 in Kreyszig's functional analysis book)

This is Problem 2.7.10 from Erwin Kreyszig's Introductory Functional Analysis with Applications. Let $C[0,1]$ denote the normed space of all (real- or complex-valued) functions defined and ...
0
votes
1answer
22 views

Show whether $N(g)=\|g\|_\infty + \|g'\|_\infty$ and $\|\cdot\|_\infty$ are equivalent on $C[0,1]$

Let $L$ be the linear subspace of $C[0,1]$ is the space of continously differentiable functions. I know I've got to show whether there exists an $a,b>0$ such that: $$a\|x\|_\infty \leq ...
1
vote
1answer
41 views

Some Norms on $V=P_n(\mathbb{R})$?

$V=P_n(\mathbb{R})$ be the vector space of all polynomials with degree $\le n$. I need to know Which of the followings are norms on $V.$ $\forall p\in V$ 1.$\|p\|^2=|p(1)|^2+\dots+|p(n+1)|^2$ ...
0
votes
2answers
26 views

Problem 8, Section 2.7 in Erwine Kryszeg's Introductory Functional Analysis With Applications

Here is Problem 8 in the Problem Set following Section 2.7 in Erwine Kryszeg's Introductory Functional Analysis With Applications: Show that the inverse $T^{-1} \colon R(T) \to X$ of a bounded ...
0
votes
0answers
22 views

Problem 2.7-9 in Erwine Kryszeg's Introductory Functional Analysis With Applications

Here is Problem 9 in the Problem Set following Section 2.7 in the book Introductory Functional Analysis With Applications by Erwine Kryszeg: Let $C[0,1]$ denote the set of all (real- or ...
3
votes
2answers
39 views

Intersection of compact convexes

Let $C_1,C_2,C_3,C_4$ be compact convexes of $\mathbb{R}^2$ such that $C_1\cap C_2\cap C_3\neq\emptyset,C_1\cap C_2\cap C_4\neq\emptyset,C_1\cap C_3\cap C_4\neq\emptyset,C_2\cap C_3\cap ...
3
votes
2answers
59 views

Metrizability of the unit ball $B_{X^*}$.

I am trying to prove the assertion: If $X$ is a separable normed space, then the unit ball in $X^*$ with the weak* topology, $(B_{X^*},\sigma(X^*,X))$, is metrizable. Firstly, I took ...
1
vote
4answers
58 views

How to exhibit the set of all the limit points of this subset of $\mathbb{R}^k$?

Let $k$ be a positive integer, let $p_0$ be a point in $\mathbb{R}^k$, let $\delta_0$ be a positive real number, and let the set $E$ be defined as follows: $$E \colon= \{ \, p\in\mathbb{R}^k \, ...
0
votes
1answer
62 views

Problem 2.7.6 in Kreyszig's Introductory Functional Analysis with Applications

Suppose that $X$ and $Y$ are two normed spaces over the same field ($\mathbb{R}$ or $\mathbb{C}$). Show that the range of a bounded linear operator $T \colon X \to Y$ need not be closed in $Y$. ...
3
votes
1answer
54 views

Closed subspace. A Hahn–Banach theorem consequence

I am trying to prove: If M is a subspace of a normed space $X$, that $\overline{M}=\bigcap\{\ker(\phi):\phi|_{M} = 0 \}$ It is really easy to see that $\overline{M} \subset ...
2
votes
0answers
36 views

Norm for a set of vectors

Let V be a normed vector space (real or complex valued) with norm $\|\cdot\|_V$. For any nonempty and bounded subset $A \subseteq V$ one can define $\|A\|$ via $$\|A\|:=\sup\{|x|:x\in A\}$$ I ...
0
votes
1answer
16 views

Product of $L_2$ norm of vectors

Is the $\sum \Vert b_k\Vert_2^2 \le\ge= \sum \Vert b_k\Vert_2^2 \Vert a_k\Vert_2^2$ ? where $b_k$ is a column vector and $a_k$ is a highly sparse row vector.
1
vote
1answer
28 views

If the dual unit ball of a normed space $X$ is metrizable in the weak-$*$ topology then $X$ is separable

Let $X$ be a normed space and $(B_{X^*},w^*)$ be the unit ball of the dual space $X^*$ endowed with the weak-$*$ topology. Here is a proof a the fact that if $(B_{X^*},w^*)$ is metrizable then $X$ is ...
3
votes
1answer
29 views

Convergence of a series of vectors in a Banach space

Let $\sum_{k=1}^\infty\lambda^{k-1}\boldsymbol{v}_k$ be a series of vectors where $\lambda^{k-1}\in\mathbb{C}$, or $\lambda^{k-1}\in\mathbb{R}$, and the $\boldsymbol{v}_k$ belong to a Banach space. I ...
1
vote
2answers
61 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 ...
1
vote
1answer
25 views

Unit ball for a special norm

What does the unit sphere for the norm on $\mathbb{R}^2$, $\displaystyle N(x,y)\rightarrow\sup_{t\in\mathbb{R}}\frac{|x+ty|}{t^2+t+1}$, look like ? My approach was to consider $y=ax$ so as to get ...
21
votes
3answers
1k views

Why are every structures I study based on Real number?

I've been studying basic concepts of inner product vector space, normed vector space and metric space. And all the inner products, norms and metrics are defined to be real-valued functions in my ...
0
votes
0answers
13 views

Problem of Closed linear transformation in Normed spaces [duplicate]

Let $X$ a normed space and let $A$ and $B$ be linear transformations such that $$X\subset D_A\rightarrow^{A} X \ \ \text{and} \ \ X\subset D_B\rightarrow^{B} X.$$ If $A$ and $B$ are closed, does it ...
1
vote
1answer
30 views

How to prove that operators are isometry on $\ell^p$/$c$?

Lately I've been studying Banach spaces and isometries, and encountered many explicit isometrys involving $c$, $c_0$, $c^*$, $c_o^*$, $\ell^1$, $\ell^\infty$, etc... ($c \subset \ell^\infty$ is the ...
2
votes
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, ...
1
vote
2answers
33 views

Is a countable union of complete subspaces complete?

I would like to ask the following, which I wanted to use a part of my proof but couldn't determine if it's right: Assume $X$ is a normed space, and $(X_n)_{n\in \mathbb N}$ complete subspaces. Must ...
1
vote
1answer
6 views

Boundedness of multivariable polynomials

How can we prove multivariable polynomials are bounded on a closed set? the boundedness theorem is for single variable functions. Does an extension theorem exist? Thank you.
1
vote
0answers
17 views

Questions regarding Banach spaces [closed]

Problem 1: Let $X$ be a Banach space and $I ∈ L(X)$ be the identity operator. Determine the action of the operator $e^I$ on $X$. Problem 2: Let $X$ be a Banach space and $A ∈ L(X)$ be a bounded ...
1
vote
2answers
33 views

Closed AND open subspaces of a normed vector space

Let $E$ be a finite dimension normed vector space. How can I show that $E$'s only both closed and open (norm-wise) subsets are $\emptyset$ and $E$ ?
0
votes
1answer
36 views

The exponential of the identity operator in a Banach space

Let $X$ be a Banach space and $I \in L(X)$ be the identity operator. Determine the action of the operator $e^I$ on $X$. Pretty stuck here, not sure exactly what it means by determine the action. ...
0
votes
0answers
20 views

Norm of linear functional in $W^{1,2}$

I want to find the norm of the following linear functional: $\phi:W^{1,2}_{[0,1]} \rightarrow\mathbb{R}$ $\phi(f)=\int^1_0t\cdot f(t)+ f^{'}(t) dt $ The norm is: ...
4
votes
2answers
69 views

Minkowski type inequality in Banach algebras

Under which circumstances it is true that $\|(A+B)^n\|^{1/n}\le \|A^n\|^{1/n}+\|B^n\|^{1/n}$ for elements $A$ and $B$ in a Banach algebra and a natural number $n$?
0
votes
1answer
67 views

Examples of double dual spaces

I am looking for examples on double dual spaces. As I know $\ell_p $ is the double dual of $\ell_p$ for $1<p,q<\infty$, $\mathcal L_p $ is the double dual for $\mathcal L_p$ for ...
0
votes
0answers
47 views

Riemann integrals of abstract functions into Banach spaces

If we define the (Riemann) integral of an abstract function, i.e. a function $f:[a,b]\to Y$ where $Y$ is a Banach space, as$$\int_a^b F(t)dt:=\lim_{\max(t_{k+1}-t_k)\to ...
2
votes
1answer
73 views

Reiterate Volterra integral operator is a contraction

I read in Kolmogorov and Fomin's Элементы теории функций и функционального анализа (p. 472 here) the statement that Volterra operator $A:L_2[a,b]\to L_2[a,b]$ defined ...
0
votes
2answers
38 views

Characterization of compact subsets in the metric space of all complex-valued sequences

Here's the statement of the Problem 4 after Section 2.5 in Introductory Functional Analysis With Applications by Erwine Kryszeg: Show that for an infinite subset $M$ in the space $s$ to be ...
0
votes
1answer
61 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 $x\in X$?
0
votes
1answer
27 views

$\ker (I-A)=\{0\}\Rightarrow\text{im }(I-A)=H$ for $A:H\to H$ compact

Let $T$ be the operator defined by $T:=I-A$ where $I:H\to H$ is the identity and $A:H\to H$ is a compact operator defined on Hilbert space $H$. In such a case, if we defined the chain of ...
1
vote
1answer
32 views

$\overline{\text{conv}}\{x_n: n\in \mathbb{N}\}$ has empty interior for a w-conv seqeunce $(x_n)_n$.

Given $x_n \to x$ weakly in a Banach space X, $\dim X = \infty$. Show that $\text{int}\left\{ \overline{\text{conv}}\{x_n:n\in\mathbb N \}\right\}$ is empty. Can anyone help me with this problem? ...
0
votes
0answers
15 views

Showing a map is a bounded linear operator.

Show that the map A : (C[0,1],∥·∥∞) → R, Ax = x(0), ∀x ∈ C[0,1] is a bounded linear operator. I know one has to show the map is continuous but I'm not sure how to go about proving it in this case. ...
1
vote
1answer
20 views

Degenerate Hilbert-Schmidt operators

Let us define a Hilbert Schmidt operator $A:L_2[a,b]\to L_2[a,b]$ by $$A\varphi:=\int_{[a,b]} K(s,t)\varphi(t)d\mu_t$$where $\mu_t$ is the linear Lebesgue measure. A degenerate case is represented by ...
2
votes
1answer
119 views

Proving that $X/M$ is a Banach space when $X$ is

I am trying to do an exercise in an introduction to functional analysis course: 1) Let $X$ be a normed space and $\{x_{n}\}_{n=1}^{\infty}\subseteq X$. Prove that $X$ is a banach space iff ...
2
votes
0answers
33 views

Proofchecking: Application of Banach-Alaoglu on weak converging nullsequence

Problem Assume $x_n \to 0$ weakly in a Banach space. Show that for all $\epsilon>0$ and for all $N\in \mathbb{N}$ there exists a $n>N$ s.t. for all $f\in X^\ast, \|f\|\leq 1$ there exists ...
0
votes
1answer
14 views

In the normed space of bounded real sequences, which subsets are closed?

I am attempting to figure out the following question. Q. Let X be the normed space of bounded real sequences and norm $\| x \|_{\infty} = Sup_{1 \leq n} |x_n|$, $x=(x_n)\in C_0$. Which of the ...
8
votes
1answer
4k views

Relations between p norms

The p-norm is given by $||x||_{p} = (\sum_{n=1}^{\infty }|x_{n}|^{p})^{1/p}$. For $0 < p < q$, it can be shown that $||x||_{p} \geq ||x||_{q}$ (1, 2). It sppears that in $R^{n}$ a number of ...
10
votes
1answer
3k views

Let $X$ be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable.

Let $X$ be an infinite dimensional Banach space. Prove that every basis of $X$ is uncountable. Can anyone help how can I solve the above problem?
3
votes
0answers
46 views

Finding L^1 centers of sets of probability distributions

Let $\mathcal{P}^n = \{ x \in \mathbb{R}^n : x \geq 0, \sum x = 1\}$. Suppose I have $p_1, \ldots, p_m \in \mathcal{P}^n$. I want to find an $L^1$ center for these points. i.e. $q \in \mathcal{P}^n$ ...
0
votes
1answer
36 views

sup norm of operator

Let $T$ be a compact linear operator defined as $$ T\circ u = \int_a^b k(x,y)\,u(y)\,dy, $$ where $k(x,y)\in C([a,b]\times[a,b])$ and $k(x,y)\ge0$ for all $x,y$, and $u\in C([a,b])$. Suppose that the ...
1
vote
0answers
25 views

Continuous functions dense in $L_p(X,\mu)$ if $X$ has a special property

Let $X$ be a metric space endowed with a measure $\mu$ satisfying the following condition: all the open and closed sets of $X$ are measurable and for any measurable set $M\subset X$ and any ...
4
votes
0answers
52 views

A normed space is not separable if and only if it contains an uncountable set of disjoint balls of the same radius

I want to show normed space $E$ is not separable iff $E$ contains uncountable set of pairwise non-intersecting balls of radius $r > 0$. Use contraposive: first prove $E$ is separable then ...
3
votes
1answer
60 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 ...
3
votes
1answer
295 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 ...