A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

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Isometric to Dual implies Hilbertable?

Let $X$ be a Banach space and suppose that $X$ is isometric to its continuous dual space $X^*$. Must $X$ be hilbertable in the sense that there exists an inner product which induces the norm on $X$? ...
12
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1answer
578 views

Are the coordinate functions of a Hamel basis for an infinite dimensional Banach space discontinuous?

The question is in the title really, but I suppose I could at least fix some notation here. Let $X$ be an infinite-dimensional Banach space - over the reals for the sake of concreteness. Use choice ...
11
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1answer
120 views

When is an operator on $\ell_1$ the dual of an operator on $c_0$?

Suppose $T:\ell_1\to\ell_1$ is a continuous linear operator. When can we say that $T$ is a dual, or adjoint, of an operator on $c_0$? In other words, under what conditions can we find a continuous ...
8
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1answer
305 views

A convergence problem in Banach spaces related to ergodic theory

Suppose $X$ is a Banach space, $T\in B(X)$, satisfied the following condition. $\sup \Big\lVert\frac{1}{n}\sum \limits_{i=0}^{n-1}T^{i}\Big\rVert<\infty$ $\frac{1}{n}\lVert ...
7
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1answer
276 views

Isomorphisms between Normed Spaces

Between two normed (linear) spaces there are several notions of isomorphisms: Linear isomorphisms: linear bijective maps Topological isomorphisms: linear homeomorphisms (due to the linearity these ...
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363 views

Is $\ell^1$ isomorphic to $L^1[0,1]$?

Can there be a continuous linear map, with a continuous inverse, from $l^{1}$ to $L^{1}(m)$ where $m$ is the Lebesgue measure on the unit interval $\left[0,1\right]?$ My thinking to this should be ...
4
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2answers
630 views

Must a weakly or weak-* convergent net be eventually bounded?

Let $\mathfrak{X}$ be a Banach space. As a standard corollary of the Principle of Uniform Boundedness, any weak-* convergent sequence in $\mathfrak{X}^*$ must be (norm) bounded. A weak-* convergent ...
3
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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, ...
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0answers
70 views

Distance from a point to a plane in normed spaces

How can we calculate the distance from a point to a plane in normed spaces ? where we are not inner product, for example: Calculate the distance in $(C[0,1],||\cdot||)$ endowed with the supremum norm ...
2
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1answer
454 views

Banach spaces and their unit sphere

Let $X$ be a normed vector space. Show that if a subsequence of a Cauchy sequence converges, then the whole sequence converges. Use the part 1 to show that $S = \{x\in X : \|x\| = 1\}$ is complete ...
10
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1answer
527 views

Liouville's theorem for Banach spaces without the Hahn-Banach theorem?

Let $B$ be a (complex) Banach space. A function $f : \mathbb{C} \to B$ is holomorphic if $\lim_{w \to z} \frac{f(w) - f(z)}{w - z}$ exists for all $z$, just as in the ordinary case where $B = ...
9
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397 views

weak* separable question

(In another question Nate Eldredge said I should ask this.) Let $X$ be a Banach space, $X^\ast$ the dual space, and $B_{X^\ast}$ the unit ball of $X^\ast$. In the weak* topology for $X^\ast$, does ...
8
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3answers
524 views

Weak convergence in reflexive Banach space

Let $X$ be a reflexive Banach space. Let $T: X \to Y$ a linear operator. I want to show that: $$T \in \mathcal{L}(X, Y) \iff ((x_n \stackrel{w}{\rightharpoonup} x) \implies (T(x_n) ...
7
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2answers
277 views

Counterexample for the stability of orthogonal projections

Let $V$ be a seperable Banach space, which is dense and continuously embedded in a Hilbert Space $H$. Let $(V_m)$ be a Galerkin scheme (See definition below) for $V$. Using the embedding we can ...
7
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1answer
405 views

Isometry between $L_\infty$ and $\ell_\infty$

It is known that there exist some isomorphism between $L_\infty$ and $\ell_\infty$, which is not explicit at all. Could someone tell me whether there exist an isometric isomorphism between ...
7
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2answers
2k views

Kernel of $T$ is closed iff $T$ is continuous

I know that for a Banach space $X$ and a linear functional $T:X\rightarrow\mathbb{R}$ in its dual $X'$ the following holds: \begin{align}T \text{ is continuous } \iff \text{Ker }T \text{ is ...
7
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1answer
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What is the predual of $L^1$

Is there a nice characterization of the predual of $L^1$? So, what does the space $X$ look like, such that $X^*=L^1$, where the star denotes the dual of a space. How do you start to find such preduals ...
6
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1answer
383 views

Rainwater theorem, convergence of nets, initial topology

I've stumbled upon a result called Rainwater's theorem a few times, it seems to be a very useful result in connection with weak convergence in Banach spaces. Rainwater's theorem. Let $X$ be a ...
6
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1answer
279 views

When can a real Banach space be made into a complex Banach space?

Suppose I have a real vector space $V$ and I would like to extend the scalar multiplication in such a way that I obtain a complex vector space. It is not difficult to see that doing so is equivalent ...
6
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3answers
618 views

Bounded Linear Mappings of Banach Spaces

This problem has been giving me some troubles. Does anyone have any ideas on how to go about proving this? Let $X$ and $Y$ be Banach spaces. If $T: X \to Y$ is a linear map such that $f \circ T \in ...
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153 views

Bounded operators with prescribed range - part II

This is a continuation of the question bellow, in a more particular case. Bounded operators with prescribed range If $X$ is a separable Banach space and $Y$ is a closed, infinite dimensional ...
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1answer
216 views

Range of bounded operator is of first category

Let $T$ be a bounded operator from a Banach Space $X$ to a normed space $Y$ such that $T$ is not onto, but $R(T)\subset Y$ is dense. Prove that $R(T)$ is of first category and not no-where dense. ...
5
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2answers
638 views

construction of a linear functional in $\mathcal{C}([0,1])$

Can someone help me to construct a linear functional in $\mathcal{C}([0,1])$ that does not attain its norm? Actually, I want to prove that $\mathcal{C}([0,1])$ is not reflexive Banach space. Is it ...
4
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1answer
118 views

Relative countable weak$^{\ast}$ compactness and sequences

I am finding serious difficulties in understanding some things about relative countable compactness and the use of sequences in proving it by my functional analysis text, Kolmogorov-Fomin's. For ...
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273 views

Dual of $l^\infty$ is not $l^1$

I know that the dual space of $l^\infty$ is not $l^1$, but I didn't understand the reason. Could you give me a example of an $x \in l^1$ such that if $y \in l^\infty$, then $ f_x(y) = ...
4
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1answer
467 views

Finding the topological complement of a finite dimensional subspace

I know that for any finite dimensional subspace $F$ of a banach space $X$, there is always a closed subspace $W$ such that $X=W\oplus F$, that is, any finite dimensional subspace of a banach space is ...
4
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1answer
219 views

Tensor products and vector valued functions

Given a non-empty set $S$ and a Banach space $X$. Let $B(S,X)$ be the space of all bounded maps from $S$ to $X$. Can we identify $B(S,X)$ with $\ell^\infty(S) \otimes X$, where $\otimes$ is some kind ...
3
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2answers
966 views

Weak limit and strong limit

Let $X$ be a Banach space and let $x_n \overbrace{\rightarrow}^w x$ and $x_n \overbrace{\rightarrow}^s z$ can we then say that $x = z$? My try: $$\| x- z\| = \sup_{\ell \leq 1} |\ell(x-z)| = ...
3
votes
1answer
410 views

What is the precise definition of predual

How does one define "predual" and the surrounding notions? More specifically: Why must there be only one predual of $X$ when $X$ is a Banach space? What is the correct notion of similarity here ...
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171 views

Bounded functionals on Banach spaces.

Let $(X, \|.\|)$ be a Banach space such that $X \subset C([0,1]) $ For every $r\in \mathbb{Q}\cap[0,1], f\rightarrow f(r)$ defines a bounded linear functional on $X$. Prove that there exists a ...
2
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340 views

fail of cantor intersection property on closed , bounded , convex sets of integrable functions

This is from my recent homework. I am asked to find a descending nested sequence of closed , bounded , nonempty convex sets $\{D_n\}$ in $L^1(\mathbb{R})$ such that the intersection is empty , where ...
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2answers
190 views

Proof of non-strictly convexity of $l_1$ and $l_{\infty}$

Given the Banach space $l_p = \left\{f\in R^n : \|f\| \leq \infty\right\}$ for $1\leq p \leq \infty$, we define the following norms $\|f\|_p = \left(\displaystyle\sum_{j=0}^{\infty} ...
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3answers
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Hilbert Space is reflexive

A normed space $X$ is reflexive iff $X^{**}=\{g_x:x\in X\}$ where $g_x$ is bounded linear functional on $X^*$ defined by $g_x(f)=f(x)$ for any $f\in X^*$. Let $X$ be a Hilbert space, would you help ...
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1answer
105 views

Stone's Theorem Integral: Bad Example!

Disclaimer This thread is just to record. See: Answer own Question It is stated as question for jeopardy. Have fun. :) Problem Given a finite Borel measure $\mu(\mathbb{R})<\infty$ and a Banach ...
0
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1answer
595 views

There are compact operators that are not norm-limits of finite-rank operators

Given an example of a Banach space for which There are compact operators that are not norm-limits of finite-rank operators. Tanks for answer
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202 views

References on relationships between different $L^p$ spaces

I am looking for detailed references containing proofs of inclusion relationships between different $L^p$ spaces and multiple counterexamples of functions in one but not the others.
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1answer
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A Hamel basis for $l^{\,p}$?

I am looking for an explicit example for a Hamel basis for $l^{\,p}$?. As we know that for a Banach space a Hamel basis has either finite or uncountably infinite cardinality and for such a basis one ...
8
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196 views

How to define Continuous $f: B\rightarrow B$ without fixed points?

Let $X$ be a infinite dimensional Banach space and $B=\{x\in X:\ \|x\|\leq 1\}$. How to define a continuous $f:B\rightarrow B$ without fixed points? Edit 1: I have changed the question a litlle, ...
7
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1answer
847 views

On every infinite-dimensional Banach space there exists a discontinuous linear functional.

On every infinite-dimensional Banach space there exists a discontinuous linear functional. Assuming the axiom of choice, every vector space has a basis. With an infinite basis, I can define on a ...
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Inclusion of $\mathbb{L}^p$ spaces, reloaded

I have a follow-up from this question. It was proved that, if $X$ is a linear subspace of $\mathbb{L}^1 (\mathbb{R})$ such that: $X$ is closed in $\mathbb{L}^1 (\mathbb{R})$; $X \subset \bigcup_{p ...
6
votes
1answer
344 views

Is kernel a complemented subspace

Let $\mathcal{A}:X\to Y$ be continuous linear operator, $X$ and $Y$ are Banach spaces. Let $\text{Im} \mathcal{A}=Y$. Is $\ker\mathcal{A}$ a complemented subspace of $X$?
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67 views

Conditions for Taylor formula

I know that, if $F:X\to Y$, where $X,Y$ are Banach spaces, is a map whose $n$-th Fréchet derivative $x\mapsto F^{(n)}(x)$ is continuous as a function of $x$ in a neighbourhood of $x_0\in X$, then the ...
5
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1answer
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If $P$ has marginals $P_1, P_2$, is $L^1(P_1) + L^1(P_2)$ closed in $L^1(P)$?

Suppose that $\mathbb{X}=\mathbb{X}_1\times \mathbb{X}_2$ and suppose that $ P$ is a probability measure on $\mathbb{X}$ with marginals $ P_i$ on $\mathbb{X}_i, i=1,2$, i.e., $$\int f_i(x_i)\, ...
5
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1answer
971 views

Proving that Tensor Product is Associative

I want to show that $X\otimes(Y\otimes Z)$ is isomorphic to $(X\otimes Y)\otimes Z$. Intuitively I think I should just choose bases $\{e_{i}\}_{i\in I}, \{f_{j}\}_{j\in J}$, and $\{g_{k}\}_{k\in K}$ ...
5
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2answers
215 views

Boundary of $L^1$ space

Is there any rigorous or heuristic notion of boundary of $L^1$ that is studied? I mean something loosely like the collection of functions or distributions defined by $$\left\{f\notin L^1: f_n\to ...
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2answers
435 views

Isomorphism of Banach spaces implies isomorphism of duals?

I can't make up my mind whether this question is trivial, or simply wrong, so i decided to ask, just in case someone sees a fallacy in my reasoning: Question: Suppose $V,W$ are two banach spaces, and ...
5
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1answer
274 views

Show sequence equicontinuous

I don't know how to prove this question: Let $X$ be a compact space, and let $(T_{n})$ be a sequence of positive linear operators on $C(X)$. Also, let $f \in C(X)$ be a strictly positive function. ...
5
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1answer
248 views

Showing that $l^p(\mathbb{N})^* \cong l^q(\mathbb{N})$

I'm reading functional analysis in the summer, and have come to this exercise, asking to show that the two spaces $l^p(\mathbb{N})^*,l^q(\mathbb{N})$ are isomorphic, that is, by showing that every $l ...
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Confused about which Hölder spaces are Banach

If $\Omega$ is an open set in $\mathbb{R}^n$, is the Hölder space $C^{k, \alpha}(\Omega)$ Banach? Or is it only that $C^{k, \alpha}(\overline{\Omega})$ is Banach, like with ordinary continuous ...
4
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1answer
297 views

Cancellation law for Minkowski sums

Let $(X,\|\cdot\|)$ be a Banach space and $A,B,C\subset X$ closed bounded non-empty convex subsets. Let $+$ denote the Minkowski symbol for addition. Does the $+$ satisfy: $$A+C\subset B+C\implies ...