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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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?
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Space of bounded continuous functions is complete

I have lecture notes with the claim $(C_b(X), \|\cdot\|_\infty)$, the space of bounded continuous functions with the sup norm is complete. The lecturer then proved two things, (i) that $f(x) = \lim ...
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Cardinality of a Hamel basis of $\ell_1(\mathbb{R})$

What is the cardinality of a Hamel basis of $\ell_1(\mathbb R)$? Is it deducible in ZFC that it is seemingly continuum? Does it follow from this that each Banach space of density $\leqslant 2^{\...
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Is it possible for a function to be in $L^p$ for only one $p$?

I'm wondering if it's possible for a function to be an $L^p$ space for only one value of $p \in [1,\infty)$ (on either a bounded domain or an unbounded domain). One can use interpolation to show that ...
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The Duals of $l^\infty$ and $L^{\infty}$

Can we identify the dual space of $l^\infty$ with another "natural space"?. If the answer yes what about $L^\infty$. By the dual space I mean the space of all continuous linear functionals.
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Inequality between $\ell^p$-norms

Suppose that a sequence $x=(x_n)$ belongs both to $\ell^p$ and $\ell^q$ ($p,q>1$, $p\neq q$). Is there any inequality between $\|x\|_p$ and $\|x\|_q$. Can one $\ell^p$ be continuously embedded into ...
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1answer
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Strong and weak convergence in $\ell^1$

Let $\ell^1$ be the space of absolutely summable real or complex sequences. Let us say that a sequence $(x_1, x_2, \ldots)$ of vectors in $\ell^1$ converges weakly to $x \in \ell^1$ if for every ...
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Nonnegative linear functionals over $l^\infty$

My purpose is a clarification of the role of the axiom of choice in constructing limits for bounded sequences. Namely, we want a linear functional of norm 1 defined on the space of all bounded complex ...
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Connections between metrics, norms and scalar products (for understanding e.g. Banach and Hilbert spaces)

I am trying to understand the differences between $$ \begin{array}{|l|l|l|} \textbf{vector space} & \textbf{general} & \textbf{+ completeness}\\\hline \text{metric}& \text{metric ...
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$C([0, 1])$ is not complete space with respect to norm $\lVert f\rVert _1 = \int_0^1 \lvert f (x) \rvert \,dx$

Consider $C([0, 1])$, the linear space of continuous complex-valued functions on the interval $[0, 1]$, with the norm $\displaystyle\lVert f\rVert_1 = \int_0^1 \lvert f(x)\rvert \,dx$. I have to ...
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Gâteaux derivative

Let $X$ be a Banach space and $\Omega \subset X$ be open. The functional $f$ has a Gâteaux derivative $g \in X'$ at $u \in \Omega$ if, $\forall h\in X,$ $$\lim_{t \rightarrow 0}[f(u+th)-f(u)- \langle ...
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How to show that this set is compact in $\ell^2$

Let $(a_n)_{n}\in\ell^2:=\ell^2(\mathbb{R})$ be a fixed sequence. Consider the subspace $$C=\{(x_n)_{n}\in\ell^2 : |x_n|\le a_n\text{ for all }n\in\mathbb{N}\}.$$ According to the book [Dunford and ...
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Prove that $X^\ast$ separable implies $X$ separable

Can someone tell me if I got the following right: Assume $X$ to be a normed vector space over $\mathbb{R}$. Prove that if the dual space $X^\ast$ is separable then $X$ is separable as well. I'm ...
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Linear functional on a Banach space is discontinuous then its nullspace is dense.

I need to prove that: If a nonzero linear functional $f$ on a Banach Space $X$ is discontinuous then the nullspace $N_f$ is dense in $X$. To prove that $N_f$ is dense, it suffices to show that $\...
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3answers
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Norm for pointwise convergence

Does there exist a norm on the space of all real-valued functions on the real line (or on an open set? a compact set?) such that convergence in this norm is equivalent to pointwise convergence?
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Compact operator maps weakly convergent sequences into strongly convergent sequences

I found the following property of compact operators in a proof, and I can't prove it. Prove that if $T \in \mathcal{L}(E,F)$ is compact, and if $u_n \rightharpoonup u$ (the sequence converges ...
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Are these two Banach spaces isometrically isomorphic?

Let $c$ denote the space of convergent sequences in $\mathbb C$, $c_0\subset c$ be the space of all sequences that converge to $0$. Given the uniform metric, both of them can be made into Banach ...
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Question about Fredholm operator

$X,Y$ are Banach spaces and $A\in B(X,Y)$ is a Fredholm operator (that is, the dimensions of ker($A$) and coker($A$) are both finite), then are closed linear subspaces ker($A$) and Im($A$) ...
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A question about Banach reflexive space

How to show that a Banach space $X$ is reflexive if its dual $X'$ is reflexive?
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If $1\leq p < \infty$ then show that $L^p([0,1])$ and $\ell_p$ are not topologically isomorphic

If $1\leq p < \infty$ then show that $L^p([0,1])$ and $\ell_p$ are not topologically isomorphic unless $p=2$. Maybe I would have to use the Rademacher's functions.
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Example of a closed subspace of a Banach space which is not complemented?

In this post, all vector spaces are assumed to be real or complex. Let $(X, ||\cdot||)$ be a Banach space, $Y \subset X$ a closed subspace. $Y$ is called $\underline{\mathrm{complemented}}$, if there ...
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Complement of $c_{0}$ in $\ell^{\infty}$

How can I show that $c_{0}$ cannot be complemented in $\ell^{\infty}$? Complement in the following sense $$c_{0}+V = \ell^{\infty}$$ And the projections are continuous.
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Show that $\liminf_{n\to \infty}x_{n}\le\alpha(x)\le\limsup_{n\to\infty}x_{n}$ for $x=(x_{n})$ in $\ell^{\infty}$

Question: Show that $\liminf_{n\to \infty}x_{n}\le\alpha(x)\le\limsup_{n\to\infty}x_{n}$ for $x=(x_{n})$ in $\ell^{\infty}$, where $\alpha$ is a bounded linear functional on $\ell^{\infty}$. I ...
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Is there an explicit isomorphism between $L^\infty[0,1]$ and $\ell^\infty$?

Is there an explicit isomorphism between $L^\infty[0,1]$ and $\ell^\infty$? In some sense, this is a follow-up to my answer to this question where the non-isomorphism between the spaces $L^r$ and $...
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3answers
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Compactness of a bounded operator $T\colon c_0 \to \ell^1$

Pitt Theorem says that any bounded linear operator $T\colon \ell^r \to \ell^p$, $1 \leq p < r < \infty$, or $T\colon c_0 \to \ell^p$ is compact. I know how to prove this in case $\ell^r \to \...
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A vector without minimum norm in a Banach space

Question: Let $E = C[0, 1]$, with sup norm. Let $K$ consist of all $f$ in $E$ such that $$\int_{0}^{\frac{1}{2}}f(s)ds-\int_{\frac{1}{2}}^{1}f(s)ds=1$$ Prove that $K$ is a closed convex subset of $E$ ...
8
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Isometric Embedding of a separable Banach Space into $\ell^{\infty}$

The problem is: Let $X$ be a separable Banach space then there is an isometric embedding from $X$ to $\ell^{\infty}$. My efforts: I showed that there is an isometry from $X^*$ (topological dual) to ...
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A compact operator is completely continuous.

I have a question. If $X$ and $Y$ are Banach spaces, we have to prove that a compact linear operator is completely continuous. A mapping $T \colon X \to Y$ is called completely continuous, if it maps ...
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An application of Riesz' Lemma

How does one prove using Riesz' Lemma that an infinite dimensional subspace $Y$ of a Banach space $X$ contains a sequence $\{x_n:n\in \mathbb{N}\}$ in the unit ball of $Y$ such that $n \neq m$ implies ...
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Direct aproach to the Closed Graph Theorem

In the context of Banach spaces, the Closed Graph Theorem and the Open Mapping Theorem are equivalent. It seems that usually one proves the Open Mapping Theorem using the Baire Category Theorem, and ...
13
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1answer
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Nested sequences of balls in a Banach space

This seems to be a fairly easy question but I'm looking for new points of view on it and was wondering if anyone might be able to help (by the way- this question does come from home-work, but I've ...
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Do continuous linear functions between Banach spaces extend?

Just wondering... Let $E$, $G$ be Banach spaces, let $U\subset E$ be a subset of $E$, and let $f:U\rightarrow G$ be a continuous linear function. Can $f$ be extended to a continuous linear function on ...
12
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Closure of the invertible operators on a Banach space

Let $E$ be a Banach space, $\mathcal B(E)$ the Banach space of linear bounded operators and $\mathcal I$ the set of all invertible linear bounded operators from $E$ to $E$. We know that $\mathcal I$ ...
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Why is $\ell^1(\mathbb{Z})$ not a $C^{*}$-algebra?

When $\ell^1(\mathbb Z)$ is equipped with the convolution as multiplication and $a^{*}_{n}=\bar{a}_{-n}$, I can prove it satisfies all conditions except $\|a^{*}a\|=\|a\|^2$, which I cannot prove nor ...
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Is any Banach space a dual space?

Let $X$ be a Banach space. Is there always a normed vector space $Y$ such that $X$ and $Y^*$ are isometric or isomorphic as topological vector spaces (that is, there exists a linear homeomorphism ...
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Banach Spaces: Uniform Integral vs. Riemann Integral

Problem Given a finite measure space $\Omega$ and a Banach space $E$. One has strict inclusion: $$\mathcal{L}_\mathfrak{U}(\mu)\subsetneq\mathcal{L}_\mathfrak{R}(\mu):\quad\int_\mathfrak{U}F\...
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Generalized Riemann Integral: Bounded Nonexample?

Reference For a convergence theorem on integral see: Riemann Integral: Uniform Convergence For an improper version of integral see: Riemann Integral: Improper Version For a comparison of integrals ...
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Equivalence of reflexive and weakly compact

In a normed space $X$ is there an equivalence between these two proposition? $1)$ $X$ is reflexive; $2)$ $B$, the unit ball of $X$, is weakly compact.
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At most finitely many (Hamel) coordinate functionals are continuous - different proof

If $X$ is a vector space over $\mathbb R$ and $B=\{x_i; i\in I\}$ is a Hamel basis for $X$, then for each $i\in I$ we have a linear functional $a_i(x)$ which assigns to $x$ the $i$-th coordinate, i.e....
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1answer
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If the dual spaces are isometrically isomorphic are the spaces isomorphic?

Let $X$, $Y$ be Banach spaces such that the duals $X^\ast$ and $Y^\ast$ are isometrically isomorphic. Are $X$ and $Y$ necessarily isomorphic? The answer to the question whether $X$ and $Y$ are ...
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$C[0,1]$ with $L^1$ norm is not Banach space.

I want to check that $(C[0,1],∥⋅∥_1)$ is not a Banach space, where $\|f\|_1 = \int_0^1 |f(x)|\,{\rm d}x$.I took $(f_n)_{n \geq 1}$ a sequence in $C[0,1]$ given by:$f_n: [0,1] \rightarrow \mathbb{R}, \...
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Example of a compact set that isn't the spectrum of an operator

This question is a follow-up to this recent question and related to that one. Is there an easy example of an (infinite-dimensional) Banach space $X$ and a non-empty compact set $K \subset \mathbb{...
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Why isometric isomorphic between Banach spaces means we can identify them?

Is the "isometric" part really necessary? For what reason is that? Eg. we prove that there is an isometric isomorphism between $(L^p)'$ and $L^q$ ($(p,q)$ conjugate) and then we identify them ...
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The space of Riemannian metrics on a given manifold.

For a finite-dimensional smooth (Hausdorff, second-countable) manifold $M$, consider the set $$\mathcal{Met}(M) = \{ g : g \text{ is a Riemannian metric on }M \}.$$ I'd like to know about the typical ...
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If weak topology and weak* topology on $X^*$ agree, must $X$ be reflexive?

Let $X$ be a Banach space and suppose that the weak topology on $X^*$ agrees with the weak* topology on $X^*$. Must $X$ be reflexive? To prove the contrapositive, it will suffice to assume that $X$ ...
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Space of Complex Measures is Banach (proof?)

How can we prove that the space of Complex Measures is Complete? with the norm of Total Variation. I have stuck on the last part of the proof where I have to prove that the limit function of a Cauchy ...
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1answer
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$C_0(X)$ is not the dual of a complete normed space

Let $X$ be any locally compact Hausdorff space and assume that it is not compact. I've heard that the Banach space $(C_0(X),\|\!\cdot\!\|_\infty)$ is not isometrically isomorphic to the (norm) dual of ...
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How do I prove the completeness of $\ell^p$?

Say $\{x_n\}$ is Cauchy in $\ell^p$ and $x$ is its pointwise limit. To argue that $x \in \ell^p$ would the following be correct: Let $\varepsilon > 0$ and let $N$ be s.t. $n,m > N$ $\Rightarrow$...
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555 views

$\ell_1$ and unconditional convergence

Thanks to the Riemann theorem we know that absolute convergence and unconditional convergence are the same for $\mathbb{R}$. In all the Frechet spaces absolute convergence implies unconditional ...
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Inclusion of $L^p$ spaces

Let $X \subset L^1(\mathbb{R})$ a closed linear subspace satisfying \begin{align} X\subset \bigcup_{p>1} L^p(\mathbb{R})\end{align} Show that $X\subset L^{p_0}(\mathbb{R})$ for some $p_0>1.$ ...