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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how to show that $c_0$ is complete

I want to show that the metric space $(c_0,d_\infty)$ is complete, where $c_0$ is the collection of all sequences $x\colon \mathbb N\to\mathbb R$ which tend to $0$. I have already shown that the space ...
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524 views

Weakly compact operators on $\ell_1$

Is the following assertion true/known? Let $V$ be a Banach space and let $T\colon \ell_1\to V$ be a bounded linear operator. Is it true that $T$ is not weakly compact if and only if there is a ...
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A Banach space of (Hamel) dimension $\kappa$ exists if and only if $\kappa^{\aleph_0}=\kappa$

A Banach space of (Hamel) dimension $\kappa$ exists if and only if $\kappa^{\aleph_0}=\kappa$. How will we prove the converse implication. One sided implication for Hilbert Space is proved in ...
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455 views

Completeness of $\ell^2$ space

I was reading up and it says that $(\ell^2,\|.\|_2)$ is complete. I know that a metric space $X$ in which every Cauchy sequence converges to an element of $X$ is called complete. And I know that a ...
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229 views

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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Surjectivity of $Id-A$ for linear operator $A$ on Banach space with $\|A\|<1$

Let $X$ be Banach space and $A:X\rightarrow X$ linear opeartor such that $\|A\|<1$. It is clear that $Id-A$ is injective. Why is it also surjective?
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210 views

Is the dual of a reflexive Banach space strictly convex?

Is the dual of a reflexive Banach space strictly convex? Why? This is a question that arouse trying to understand the theory behind approximation by finite element methods.
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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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Proving that a Hölder space is a Banach space

I am trying to show that the Hölder space $C^{k,\gamma}(\bar{U})$ is a Banach space. To do this, I successfully proved that the mapping $\| \quad \| : C^{k,\gamma}(\bar{U}) \to [0,\infty)$ is a norm, ...
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Differentiability: Partially Defined Functions

Reference These ideas came to my mind while reading Lee's Introduction to Smooth Manifolds (cf. p. 45). Definition Let $E$ and $F$ be two Banach spaces together with a plain subset $A\subseteq E$. ...
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Is the complex Banach space $C([0,1])$ dual to any Banach Space?

I've been able to show that the extreme points of $C([0,1])$ are the continuous functions that take values on the unit circle. However, I'm not sure how to reason from here as to whether or not it is ...
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Bounded linear operator maps norm-bounded, closed sets to closed sets. Implies closed range?

Question Suppose $T:X\rightarrow Y$ is a continuous, injective linear operator between Banach spaces. Suppose, in addition, that $T$ maps norm bounded closed sets in $X$ to closed sets in $Y$. Then ...
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368 views

Linear isometry between $c_0$ and $c$

The following question is an exercise and so I'm just looking for advices and not for answers if it's possible. I have the following sets in $l^\infty$ $$c_0 := \{x_n \in l^\infty: \lim x_n = 0\} ...
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A certain element which makes functionals positive

Suppose $X$ is a possibly non-separable Banach space and let $X^*$ be its dual. Also, let $(f_n)_{n=1}^\infty$ be a sequence of [EDIT: linearly independent] norm-one functionals in $X^*$. Does there ...
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Space of $\mathbb{R}$ valued sequences converging to $0$. Some basic results.

Let $C_0(\mathbb{R})$ be the space of $\mathbb{R}$ valued sequences converging to $0$. Let $l_n$ be a positive sequence in $\mathbb{R}$ such that $\sum\limits_{n=1}^\infty l_n=1$. We define $$ ...
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1answer
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Approximation Property: Characterization

As reference the german wiki: Approximationseigenschaft Problem Given a Banach space. Suppose it has the approximation property: $$C\in\mathcal{C}:\quad\|T_N-1\|_C\to0\quad(T_N\in\mathcal{F}(E))$$ ...
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Stone's Theorem Integral: Avanced Integral

Reference This problem grew out from: Stone's Theorem Integral: Basic Integral Problem Given the real line as measure space $\mathbb{R}$ and a Hilbert space $\mathcal{H}$. Consider a strongly ...
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289 views

Book suggestion geometry of Banach spaces

I am studying geometry of Banach spaces and applications in metric fixed point theory. Does anyone have a good recommendation of books//lectures/resources/etc.? Thanks.
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Let $V:= (C([0,1]),\|\cdot\|$) with $\|f\|:= \int_0^1|f(x)| \ dx$. Consider the function $f_n$ and show that V is not a Banach Space.

The Assignment: Let $V:= (C([0,1]),\|\cdot\|$) with $\|f\|:= \int_0^1|f(x)| \ dx$. Consider the continuous function $f$ for all $n \in \mathbb{N}$: $$f_n: [0,1] \rightarrow \mathbb{R}, \ ...
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119 views

Can a vector space be complete for two non-compatible norms?

Let $V$ be a vector space and suppose that we have two non-compatible norms on it, i.e. I distinguish $E = (V, \|\cdot\|_1)$ from $F = (V, \|\cdot\|_2)$ and I ask that $\not\exists C>0 \; ...
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141 views

a compact operator on $l^2$ defined by an infinite matrix

Let $A$ be an infinite matrix such that $\displaystyle \sum_{i,j}|a_{i,j}|^2<\infty$. Then $A$ defined a compact operator on $l^2$.
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1answer
229 views

Subspace isomorphic to a complemented subspace

I'll begin by writing down the definitions I'm using, to avoid confusion. Let $X$ be a Banach space and let $Y$ be a subspace of $X$. We say that $Y$ is complemented in $X$ if there exists a linear ...
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Show $T$ is invertible if $T'$ is invertible where $T\in B(X)$, $T'\in B(X')$

Seems simple enough but I can't quite get it. $X$ is a complex Banach space, and $T\in B(X)$, $T'\in B(X')$ is its adjoint. Suppose $T'$ is invertible. How can we show that $T$ is invertible? I have ...
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A question about Banach limits

I'm trying to prove that there exists a multiplicative linear functional in $\ell_\infty^*$ that extends the limit funcional that is defined in $c$ (i.e., im looking for a linear functional $f \colon ...
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there is $M<\infty$ such that $\sum_{n} |\hat{f}(n)|\le M\int_{0}^{2\pi}|f(t)|dt$ for each $f\in X$

for $f\in L^1[0,2\pi]$ define $$\hat{f}(n)=\int_{0}^{2\pi} f(t)e^{-int} dt$$ for $n\in\mathbb{Z}$, $X$ is a closed linear subspace of $L^1[0,2\pi]$ such that $\sum_{n} |\hat{f}(n)|<\infty$ for each ...
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317 views

Quotient space of $l^p$ that isometrically isomorphic to $l^p$

Let $Y=\{x\in l^p:x(2n)=0\}$, $1\leq p \leq \infty$. It can be proved that $Y$ is closed subspace of $l^p$. Define the quotient space $l^p/Y=\{x+Y:x\in l^p\}$. Then, by the fact that $Y$ closed, a ...
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34 views

Embedding: Extension

Problem Given Banach spaces $E_0$ and $E$ Regard dense domain: $$\overline{\mathcal{D}_0}=E_0\quad\overline{D}=E$$ Consider an embedding: ...
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$Tf=\sum\limits_{n=1}^\infty f(n)x_n$ is surjective from $\ell^1$ to a separable Banach space

Let $X$ be a separable Banach space and let $\mu$ be a counting measure on $\mathbb{N}$. Suppose that $\{x_n\}_{n=1}^\infty$ is a countable dense subset of the unit ball of $X$ and define $T: ...
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Bochner Integral: Axioms

Disclaimer It is meant to record. See: Answer own Question It is written as question. Have fun! :) Reference It is taken from the original paper: S. Bochner, Integration It is related to: Bochner ...
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524 views

Spectrum of a nilpotent operator

Let $X$ be a Banach space and $A:X\to X$ be a bounded operator such that $A^n=0$ for some $n\in \mathbb{N}$. Is the spectrum of $A$ finite, countable ?
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Is the complement of an open ball in a Banach space connected?

Let $B$ be a real Banach Space whose dimension is at least $2$, and let $S$ be a subset of $B$ that is an open ball. Is the complement of $S$ (with respect to $B$) always connected? Idea One could ...
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885 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 ...
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Characterisation of norm convergence

Let $X$ be a Hilbert space and $(x_n)\in X^{\mathbb N}$ be a sequence. Then the following statements are equivalent (with $x \in X$): We have $x_n \to x$, i.e. $\| x_n -x \| \to 0$ and we have $x_n ...
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$Conv(Ex((C(X))_1))$ is dense in $(C(X))_1$?

Let $X$ be compact Hausdorff, and $C(X)$ the space of continuous functions over $X$. Denote the closed unit ball in $C(X)$ by $(C(X))_1$, then it can be shown $f$ is an extreme point of $(C(X))_1$ if ...
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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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Representation of the elements of a normed vector space that has a dense subset.

Suppose that $U$ is a normed vector space and $S\subset U$ is dense.prove that every element of $U $ can be written as an absolutely convergent series of the finite linear combination of the ...
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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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44 views

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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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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The space $C[a,b]$ with $\|f\|:=\int_a^b |f(x)|d(x)$ is a normed space but not a Banach space

Let $-\infty\lt a\lt b \lt \infty$. Let $C[a,b]$ is space of continuous functions and a function $\|.\|:C[a,b]\to R$ is given by $\|f\|:=\int_a^b |f(x)|d(x)$ a). Show that $\|\cdot\|$ is a norm in ...
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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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1answer
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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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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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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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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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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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338 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 ...