3
votes
1answer
49 views

Vector spaces isomorphic, then dual spaces isomorphic

If we know that there is a (topological) isomorphism between two Banach spaces $X,Y$ called $\phi \in L(X,Y)$. Then the appropriate isomorphism between the dual spaces $X',Y'$ is given by $\phi' \in ...
3
votes
0answers
38 views

The dual of the Annihilator

Let $X$ be a Banach space, and $I$ be a closed subspace. Then it's known that $(X/I)^*=I^{\perp}$. My question is what is the second dual of $X/I$? or what is the dual of $I^{\perp}$ ? If we know ...
1
vote
1answer
27 views

Continuity of an application between function spaces.

I'm trying to prove the following statement... Let $f:[a,b] \times \mathbb{R} \to \mathbb{R}$ a bounded and continuous function, $t_{0} \in [a,b]$, $x_{0} \in \mathbb{R}$, $r>0$ and $$B= \{ x ...
2
votes
1answer
33 views

$f_{n}$ converges in $L^{1}$ and $\|g_{n}\|_{L^{1}(\mathbb R)}\leq \|f_{n}\|_{L^{1}(\mathbb R)}\implies {g_{n}}$ converges in $L^{1}(\mathbb R)$?

Let $f_{n}, g_{n}\in L^{1}(\mathbb R)$ (Lebesgue space). Suppose there exist $f\in L^{1}(\mathbb R)$ such that $\|f_{n}-f\|_{L^{1}(\mathbb R)}\to 0$ as $n\to \infty;$ that is, the sequence $\{f_{n}\}$ ...
5
votes
1answer
55 views

Closure of compact sets in Banach space

Let $(X,\vert\vert\cdot\vert\vert)$ be a Banach space. For each $k\in\mathbb{N}$ let $A_k\subseteq X$ be compact and $r_k\in\mathbb{R},r_k>0$, such that $$A_{k+1}\subseteq \{x+u\vert x\in A_k ...
0
votes
1answer
21 views

understanding a proof involving equivalence of norms in finite dim. linear normed spaces

I am reading the proof of the theorem shown below (from Linear Functional Analysis by Rynne and Youngson). I can't figure out why the part I highlighted in red is true. I understand why $S$ is compact ...
0
votes
1answer
53 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 $D^\alpha f$ ...
1
vote
1answer
24 views

Bounded operators on unit ball and equicontinuity

Let $X$ be a Banach space and $B$ be the closed unit ball contained in $X$. Let $\{T_{\alpha}\}$ be a family of bounded linear operators from $B$ to $V$, a normed vector space. My question is: Suppose ...
0
votes
0answers
25 views

Is the set, $\{f\in Y: \left\|f\right\|_{Y} \leq C \ \text{and} \ \left\|(|f|)\right\|_{X}\leq C \}$, closed in $(Y, ||\cdot||_{Y})$?

Put, $X= \text{The space of "nice" complex valued functions on } \mathbb R; \text{that is}, f:\mathbb R \to \mathbb C;$ so that $X$ is Banach space with respect to the norm ...
0
votes
1answer
34 views

when one can expect $\left \| |x|^{2}x- |y|^{2}y \right \| \leq \frac{1}{2} \left \| (x-y) \right \|$?

Suppose $(X, \left\|\cdot \right \|)$ is Banach algebra with the property that $\left\| (|x|^{2}x) \right\| \leq \left\|x\right \|^{3};$ for every $x\in X.$ Let $y_{0}\neq 0 \in X$ and fix it; and ...
1
vote
1answer
37 views

Proof check for completeness

I'd like to see if the proof I have is adequate. Statement. Let $X$ and $Y$ be Banach space, the product $X\times Y$ is a vector space under coordinate operations with norm $$ \|(x,y)\| = \|x\|_X ...
4
votes
1answer
95 views

Homeomorphisms between infinite-dimensional Banach spaces and their spheres

As I know Cz. Bessaga has proved that an infinite-dimensional Banach space is homeomorphic to its unit sphere. Unfortunately I do not have his book but I want to know is this theorem true without ...
3
votes
2answers
79 views

Why the image of a linear map is not always a Banach space?

I have a question: Let's think about the map $T:V \rightarrow \text{ran}(T)$ and $V$ be a Banach space. Then we have that this is the same as the quotient map $[T]:V \rightarrow V/\ker(T)$ where the ...
1
vote
1answer
40 views

norm of Frechet derivative in point.

Let $ E = \mathcal B([0,1], \Bbb R) $ with supremum norm. Now I can define function $ F:E \ni f \rightarrow ||f||^2 - f(0) \in \Bbb R$ My task: 1)Show the differentiability of $F$ in: $ f_0: ...
1
vote
1answer
34 views

differences between Banach spaces and $\Bbb R^n$.

Can you please tell me, what are the biggest differences between Banach spaces and $\Bbb R ^n$? I am trying to understand the Frechet derivative.
2
votes
1answer
56 views

Hyperplane which can separate two closed convex set

Define: $$E=:\ell^1(R)=\{x=(x_n)_n: \|x\|_1=\sum_{n=1}^{\infty}|x_n|<\infty\}$$ We have two closed convex sets $X,Y$ as subsets of normed vector space $\ell^1(R)$ with $X\cap Y=\emptyset$ and $Y$ ...
0
votes
1answer
45 views

Check in which points is f differentiable.

Check in which points is f differentiable: $f: l_{ \Bbb R}^{ \infty} \ni x \rightarrow ||x||_{ \infty} \in \Bbb R$ I started with this: $ \lim_{||h||_{\infty} \to 0} ...
0
votes
1answer
38 views

Show that the following functional is Frechet differentiable in Hilbert space

I need to show that the following functional is Frechet differentiable: $$ f(u) = \|u\|^2_{H} \ \ \text{in a real Hilbert space} \ \ H $$ Solution: As far as I understand, I need to take a Taylor ...
1
vote
1answer
71 views

Show that the following functional is Frechet differentiable

I am new to this and I need to show that the following functional is Frechet differentiable: \begin{equation} f(u) = \sin(u(1)) \ \ in \ \ C[0,1] \end{equation} What I have already done: ...
3
votes
2answers
65 views

Does $L^1$ contain a subspace isomorphic to $c_0$?

Can any $L^1$ space, say $L^1(\mathbb{R})$, have some subspace isomorphic to $c_0$? I guess not but I don't see an argument right now.
1
vote
0answers
53 views

$L^{\infty}$ is p-concave

I want to show that the $L^{\infty}$ on a Banach lattice $X$ is $p$-concave with $M_{(p)}(L^{\infty})=1$. Where $L^\infty=L^\infty(X,\mathcal{M},\mu)$. Recall that a Banach lattice $X$ is said ...
1
vote
1answer
46 views

Absolutely convergent sums in Banach spaces

Let's say a sum of elements in a Banach space is absolutely convergent if even the sum of the norms converges, i.e. $\sum_{i=1}^\infty ||x_i|| \le \infty$. This condition implies that the sum of the ...
0
votes
1answer
180 views

Quotient norm and actual norm

I have a question about the proof that $X\backslash U$ is a Banachspace if $X$ is one and $U$ is closed. In my book it is said, that for $x_k \in X$ and a series $\sum_{k=1}^{\infty}||[x_k]||< ...
0
votes
0answers
76 views

Application of fixed point theorem for an integral equation.

Consider the following non-linear integral equation: $f(x)=\lambda \int_a^b K(x,y;f(y))dy + \phi(x)$ with $K$ and $\phi$ continuous functions , such that $K$ satisfies the Lipschitz condition on the ...
0
votes
1answer
54 views

Proof of the nonexistence of an identity $\phi$ involving convolution

The Banach space $L^1(\mathbb{R}^n)$ is an algebra with a product (convolution) which is both commutative and associative. But this algebra does not have a multiplicative identity. An attempt to show ...
0
votes
1answer
98 views

Let $E$ be a Banach space, prove that the sum of two closed subspaces is closed if one is finite dimensional

Let $E$ be a Banach space and let $S$ and $T$ be closed subspaces, with dim$\space T<\infty$. Prove that $S+T$ is closed. To prove that $S+T$ is closed I have to show that for any limit point $x$ ...
0
votes
1answer
40 views

Limit of a sequence that is Cauchy with respect to a family of seminorms

Definition: Let $p$ be a seminorm on some space $X$. A sequence $(x_n)$ in $X$ is said to be Cauchy with respect to $p$ if for any $\epsilon > 0$, there is an $N > 0$ such that $p(x_m - x_n) ...
1
vote
3answers
205 views

From analysis of realvalued functions to analysis of Hilbert/Banach-valued functions

Does anybody know of a text (doesn't matter which form: article, book etc. - anything's welcome) in which it is described which result from real analysis also hold for Hilbert/Banach spaces ? I'm ...
0
votes
1answer
121 views

Derivative Bilinear map

I wanted to calculate the derivative of a continuous bilinear map $B: X_1 \times X_2 \rightarrow Y$. (Does anyhere know whether there is a generalisation of the notation $L(X,Y)$ that you use for the ...
1
vote
1answer
102 views

Reflexivity of a Banach space without the James map

The reflexivity of a Banach space is usually defined as having to be enforced by a particular isometric isomorphism. Namely the map that takes each element to the evaluation, which is already an ...
0
votes
0answers
87 views

Questions about the Gateaux derivative

We defined that a function is Gateaux differentiable, if all directional derivatives exist. I just wanted to check, whether I got a few things right: Now I wanted to ask, whether it is true that if ...
3
votes
2answers
750 views

Prove that $C^1([a,b])$ with the $C^1$- norm is a Banach Space

Consider the space of continuously differentiable functions, $$C^1([a,b]) = \{f:[a,b]\rightarrow \mathbb{R}|f,f' \text{are continuous}\}$$ with the $C^1$-norm $$||f|| = \sup_{a\leq x\leq ...
1
vote
2answers
320 views

Understanding the Frechet derivative

What is the relationship between the normal 'high school' concept of a derivative and a Frechet derivative? According to wikipedia the Frechet "extends the idea of the derivative from real-valued ...
4
votes
1answer
79 views

The Banach space $c_0$ is $C^{\infty}$-smooth.

In this paper, J. Eells defines this notion of $C^r$-smoothness for Banach spaces: A Banach space $E$ is $C^r$-smooth, $r \geq 0$, if there exists a nontrivial (that is, nonzero) $C^r$ function ...
2
votes
1answer
171 views

annihilator of an intersection in infinite dimension

Given two subspaces of an infinite dimensional Banach space, is the sum of their annihilators dense in the annihilator of their intersection?
2
votes
2answers
144 views

Strong convergence of operators

I'm working through the functional analysis book by Milman, Eidelman, and Tsolomitis, and I have a question. The book states a lemma that I'm a bit confused about: A sequence of operators $T_n\in ...
5
votes
1answer
62 views

A basic question on Type and Cotype theory

I'm studying basic theory of type and cotype of banach spaces, and I have a simple question. I'm using the definition using averages. All Banach spaces have type 1, that was easy to prove, using the ...
1
vote
1answer
28 views

$K(t,x)=\inf_{x=a+b,\ a\in X,\ b\in Y}\{\|a\|_X+t\|b\|_Y\}$ and $\int_0^\infty \frac{|K(t,x)|^p}{t}dt<\infty$ implies $x=0$?

Let $X,Y$ be two Banach spaces with respective norms $\|\cdot\|_X$ and $\|\cdot\|_Y$. Suppose that $X$ and $Y$ are subsets of a vector space $Z$. Define $K(t,x)$ for $t\in (0,\infty)$ and $x\in X+Y$ ...
2
votes
2answers
53 views

Geometry of the space $C[a, b]$ respect to the norm $\lvert \lvert x \rvert\rvert_{\infty} = \max_{t\in [a.b]}\lvert x(t)\rvert$.

I have studied that the space $C[a, b]$ of all scalar-valued (real or complex) continuous functions defined on [a, b] is a Banach space with respect to the norm $\lvert \lvert x \rvert\rvert_{\infty} ...
3
votes
3answers
110 views

How to prove that sequence spaces $l^{p}, l^{\infty}$ and function space $C[a. b]$ are of infinite dimension

I am studying about the sequence space $l^{p}, l^{\infty}$ and function space $C[a. b]$. It is mentioned in the book that all of these spaces are of infinite dimension. I want to prove that these ...
3
votes
4answers
183 views

Show that $c$ is closed in $l^{\infty}$

Let $$c=\{ (a_i)_{i \in \mathbb{N}} \ ; \ \ a_i \in \mathbb{R}\ ,\ \forall i \in \mathbb{N} \ , \ \mbox{exist} \ \displaystyle \lim_{i \to \infty}(a_i)\}$$ $$l^{\infty}=\{ (a_i)_{i \in \mathbb{N}} \ ...
4
votes
0answers
90 views

Differential calculus on Banach space

I'm revising for my upcoming test, and this problem dated back some years ago. I've been working on this problem for almost a day, but I don't even know how to start it correctly. Problem Given the ...
1
vote
1answer
32 views

Is there any space with normal structure but not uniform normal structure?

It is clear that uniformly normal structure implies normal structure. But, is there any space that has normal structure but it doesnt have uniformly normal structure?? Would you mind to give me some ...
0
votes
0answers
26 views

Lemma 3.2 from “Positive solutions for third order semipositone boundary value problems”

How de prove this lemma please : Assume that: $w(t)$ is nondercreasing and $w(t)>0$ on $(q,1]$ , $\frac12<p<q<1$ hods . Let $z\in C^2[0,1]\cap C^3(0,1)$ satisfy $z'''(t)\geq 0$ 0n ...
0
votes
1answer
87 views

Sum of Banach spaces

Let $H^2(\mathbb{R}^3)$ the usual Sobolev space and consider the following set $$X=\bigg\{u\bigg|u=\phi+\frac{Q}{|x|},\phi\in H^2,\,\, Q\in\mathbb{C}\bigg\}$$ I observe that the decomposition is ...
3
votes
1answer
469 views

Unit ball of a Separable Banach Spaces is metrizable

Please help me to understand why the unit ball of a separable banach space is metrizable, when it is given the induced topology from the weak topology. Specifically, my problem is I think I'd be able ...
1
vote
1answer
131 views

Strong and weak-* convergence for bounded linear maps

The textbook I am using says that a sequence $(T_n)$ in $\mathcal{B}(X,Y)$ for $X$, $Y$ normed linear spaces converges strongly to $T$ if $\lim_{n\rightarrow\infty}T_nx=Tx$ for every $x\in X$. The ...
1
vote
1answer
135 views

Banach Space continuous function

On the Banach space $(C([-1,1]), ||\cdot||_\infty ) $ consider the operator given by $(Tf)(x)= \dfrac{1}{3} \displaystyle\int^1_0txf(t)\ dt + e^x - \dfrac{\pi}{3} $ 1) prove that the mapping is a ...
3
votes
1answer
56 views

Evaluating difficult spectrum

Can anyone see how to show the spectrum of the bounded linear operator $T$ on $l^1$ defined by $$T((\alpha_j)) = (\alpha_j - 2\alpha_{j+1} + \alpha_{j+2})$$ is the cardioid $$\{(r, θ) : 0 ≤ θ < 2π, ...
1
vote
0answers
72 views

(Real Analysis) Integration of two functions

Note ($\Omega,A,\mu)$ is a finite additive space. Let $f\in \bar S(A, \mathbb{R})$ and $y\in Y$ where $Y$ is Banach space. Prove $\int _\Omega yf d\mu = y \int _\Omega f d\mu$. Also, for any $X\in A$, ...