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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Proof of Eberlein–Smulian Theorem for a reflexive Banach spaces

Looking for the proof of Eberlein-Smulian Theorem. Searching for the proof is what I break with this morning. Some of my friends recommend Haim Brezis (Functional Analysis, Sobolev Spaces and ...
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Hamel Basis in Infinite dimensional Banach Space without Baire Category Theorem

Prove that every Hamel basis in an infinite dimensional Banach Space is uncountable without using Baire Category Theory. We are assuming axiom that vector space dimension (if exist) is well-defined. ...
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Dense subsets in tensor products of Banach spaces

Assume $B_1$ and $B_2$ are Banach spaces of univariate functions. Moreover, assume that the sets $D_1 \subset B_1$ and $D_2 \subset B_2$ are dense with respect to the respective norms ...
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Semigroups: Entire Elements (II)

Problem Given a Banach space $E$. Consider a contraction C0-group: $$T:\mathbb{R}\to\mathcal{B}(E):\quad\|T(t)x\|\leq\|x\|$$ Define its generator by: $$Ax:=\lim_{h\to0}\frac{1}{h}(T(h)x-x)\in E$$ ...
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Generalized Riemann Integral: Uniform Convergence

Disclaimer This thread is meant to record. See: Answer own Question And it is written as question. Have fun! :) Reference For a bounded nonexample of integrability see: Riemann Integral: Bounded ...
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Group of operators such that $|T(t)x|\geq c |x|$

Let $X$ be a Banach space. Can I have an example of a strongly continuous group of operators $T(t)$ such that $$|T(t)x|\geq c |x|, \ t\in\mathbb{R}$$with $c>1$. For $c=1$, I know examples of ...
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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} ...
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249 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 $x$ ...
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Dual Space of Convergent Sequences

For every linear functional $T$ on the space of convergent sequences in $\mathbb{R}$, how can I show it can be expressed $T(\{s_{n}\}) = \sum_{n \in \mathbb{N}} s_{n}T(e_{n})$ where $e_{n}$ are the ...
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Finding a condition to make a map a contracting map

Consider $(\mathbb{C}^n ,||\cdot||_{1})$ where $||x||_{1} = \sum\limits_{i=1}^{n} |x_{i}|$ and $d_{1}(x,y) = ||x-y||_{1} = \sum\limits_{i=1}^n |x_{i} - y_{i}|$.
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Does the open mapping theorem imply the Baire category theorem?

A nice observation by C.E. Blair1, 2, 3 shows that the Baire category theorem for complete metric spaces is equivalent to the axiom of (countable) dependent choice. On the other hand, the three ...
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Norms on C[0, 1] inducing the same topology as the sup norm

This is an old homework problem of mine that I was never able to solve. The solution may or may not involve the Baire category theorem, which I am terrible at applying. Let $C[0, 1]$ denote the ...
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Reinventing The Wheel - Part 2: The Lebesgue Integral

Disclaimer After struggling for some time to find an appropriate definition for the notion of integration I came across another attempt for which I would need your help deciding to what extend this ...
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302 views

Are there spaces “smaller” than $c_0$ whose dual is $\ell^1$?

It is well known that both the sequence spaces $c$ and $c_0$ have duals which are isometrically isomorphic to $\ell^1$. Now, $c_0$ is a subspace of $c$. My question - is there an even smaller subspace ...
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Vector, Hilbert, Banach, Sobolev spaces

Trying to wrap my head around all these different spaces. Which one is the most general? Can you summarize the differences between them? Is there a notable space that I missed?
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A few questions about the Hilbert triple/Gelfand triple

I am attempting to fully understand Hilbert triples by reading Brezis' Function Analysis book. Consider $V \subset H \subset V^*$, where $V$ is Banach and $H$ is Hilbert. $V$ is dense in $H$. ...
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On the density of $C[0,1]$ in the space $L^{\infty}[0,1]$

It's easy to show $C[0,1]$ is not dense in $L^{\infty}[0,1]$ in the norm topology, but $C[0,1]$ is dense in $L^{\infty}[0,1]$ in the weak*-topology when take $L^{\infty}$ as the dual of $L^{1}$. how ...
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A property of exponential of operators

Let $X$ be a Banach space. $A\in B(X)$ is a bounded operator. we can define $e^{tA}$ by $$e^{tA}=\sum_{k=0}^{+\infty}\frac{t^kA^k}{k!}$$ I am interested in this property: If $x\in X$, such that the ...
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Isomorphic embedding of $L^{p}(\Omega)$ into $L^{p}(\Omega \times \Omega)$?

Let $(\Omega,\mu)$ be a finite measure space such that $\mu(\Omega)=1$. Suppose $1\leq p \leq \infty$. Let $\psi \colon L^p(\Omega) \to L^p(\Omega \times \Omega)$ be the map which maps $f$ onto the ...
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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 ...
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Infinite dimensional constant rank theorem

Suppose you have an analytic map $\phi : E \rightarrow \mathbb{C}^n$, where $E$ is a complex Banach space, and such that the rank of $D \phi$ is constant. Is it true then that the set ...
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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 $X$? ...
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How to deduce open mapping theorem from closed graph theorem?

These two theorems are equivalent but I can not figure out how to deduce the open mapping from the closed graph. Can anyone give a hint or some reference?
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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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Why is this inclusion of dual of Banach spaces wrong?

Ive been struggling the last days on this paradox, please I need help! Let $$E\subset F$$ be two Banach spaces equipped with the same norm. Some people told me that $$F^* \subset E^*$$ with $E^*$ ...
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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 ...
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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 ...
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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 ...
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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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 = ...
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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 ...
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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) ...
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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 ...
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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 ...
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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 ...
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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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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 ...
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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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What makes compact operators special?

I would like to understand why compact operators are considered so special to consider them as an extra class of operators. Over Hilbert spaces these (as far as I know) these are the ones with ...
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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. ...
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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 ...
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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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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 ...
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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 ...
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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 ...