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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Properties of a $B^\ast$-algebra

Defining a complex Banach algebra as a $B^\ast$-algebra when it is equipped with an application $\ast:B\to B$ such that for any $x,y\in B$ $$(x+y)^\ast=x^\ast+y^\ast,\quad(xy)^\ast=x^\ast ...
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Convergence of sequence of functions on Banach space

Let $\{f_{\alpha_n}\}\subset{\cal L}_2^0(\mathbb R)$ be a sequence function converging to $g$ where ${\cal L}_2^0(\mathbb R)$ is a Banach space defined by $$ {\cal L}_2^0(\mathbb ...
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Norm of a character in a non-unital Banach algebra

Let $\cal A$ be an abelian non-unital Banach algebra and $h:{\cal A}\to {\Bbb C}$ be a homomorphism. If ${\cal A}$ has an approximate identity $\{e_i\}$ such that $||e_i||\leq 1$ for all i, then ...
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Strictly singular surjections

Each separable Banach space $E$ is a quotient of $\ell_1$. In particular, if $E$ does not contain a subspace isomorphic to $\ell_1$ then every such a surjection is strictly singular. My question is ...
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Complex Measure Integration

Disclaimer: This thread is meant as record and written in Q&A style. Additional answers are heartly welcome, too! Integration w.r.t. complex measure usually is defined via the Radon-Nikodym ...
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Spectral Measure Integration: Unbounded Functions?

Given a Hilbert space $\mathcal{H}$ and spectral a measure $E:\Sigma(\Omega)\to\mathcal{B}(\mathcal{H})$. How to define the integral for unbounded measurable functions: ...
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When Heine - Borel theorem holds

If $\cal A$ is an abelian Banach algebra, then its maximal ideal space $\Omega$ is a compact Hausdorff space. In the proof of this theorem, the author says, since $\Omega \subset ball \cal A^*$, it ...
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isometric isomorphism between normed spaces and its dual

Let $E$ and $F$ be normed spaces. If $E \equiv F$ (isometry isomorphic), Does $E^* \equiv F^*$ (isometry isomorphic)? Where $E^*$ and $F^*$ are continuous dual spaces.
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Every homomorphism on a C*-algebra is a *-homomorphism

The following is a proposition of Conway's Functional analysis: and also he uses below exercise to proof above proposition: But I do not know how he uses the Exercise and say $||h||=1$. Please ...
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Example of a proper, dense ideal in a unital Banach algebra

I know that if ${\cal A}$ is a non- unital Banach algebra, then ${\cal A}$ may contain some proper, dense ideals. I need an example of a proper, dense ideal in a unital Banach algebra or show that it ...
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Examples of compact sets that are infinite dimensional and not bounded

In an infinite dimensional Banach space, does a compact subset have to be finite dimensional? I know it cannot contain any infinite dimensional balls, if this mean it has to be finite dimensional, ...
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Why are “not bounded” operators not everywhere defined?

Let $X, Y$ be Banach spaces, $\mathcal{D}(T)$ a subspace of $X$, and $T\colon X\to Y$ a linear map. Such a $T$ is commonly called an unbounded linear operator, where unbounded just means that the ...
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Bochner: Lebesgue Obsolete?

Bochner's notion of integral: $$F\text{ Bochner integrable}:\iff \exists S_n\in\mathcal{S}:\quad \int\|S_m-S_n\|\mathrm{d}\mu\to 0\quad(S_n\to F)$$ This version totally circumvents Lebesgue's notion ...
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Understanding the proof that $c_0$ is a closed subspace of $\ell^\infty$

The problem is given: source Let $c_0$ be a space of real sequences $x = \{x_n\}_{n = 1}^\infty=0$ converging to $0$. Let $\ell^\infty$ be a set of real sequences $w = \{w_k\}^\infty_{k=0}$ ...
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tough lp inequalities

Let $1<p<\infty$. If possible, find a positive decreasing sequence $w_1>w_2>\cdots$ such that $\lim w_i=0$, and a (uniform) constant $K>0$, such that ...
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Direct sum decomposition of vector spaces and their tensor powers

Let $V$ be a locally convex vector space and let $U$ be a finite-dimensional subspace of $V$. The Hahn-Banach theorem guarantees that there exists a closed subspace $W$ of $V$ such that $$V=U\oplus ...
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Quotient of a Banach space $X$ gets quotient topology under standard norm induced from $X$.

Let $X$ be a Banach space and $Y$ be a closed subspace of $X$. Then there is a canonical norm on $X/Y$. I want to show that this norm induces quotient topology on $X/Y$. Any hint/solution? I was ...
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26 views

Show that an operator is weakly compact

If $(X,\Omega,\mu)$ is a finite measure space, $k\in L^\infty(X\times X, \Omega\times \Omega,\mu \times \mu)$ , and $K:L^1(\mu)\to L^1(\mu)$ is defined by $$(Kf)(x)=\int k(x,y) f(y) d\mu(y)$$ show ...
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Weakly compact operator on $c_0$ is compact

Show that if $T\in {\cal B}(c_0)$ and $T$ is weakly compact, then $T$ is compact. My attempt: $T$ is weakly compact, so there is a reflexive space $X$ , and operators $A\in {\cal B}(X,c_0) $ and $B ...
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Continuity of a linear map on Banach space.

A function $f$ on $X$ is said to separate points of $X$ if for $x_1\neq x_2$ then $f(x_1)\neq f(x_2)$. Suppose $A$ is a subset of $Y^{*}$, the space of functionals on a Banach space $Y$, which ...
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Dimension for a closed subspace of $C[0,1]$.

Let $X \subset C^1[0,1]$ be a closed subspace of $C[0,1]$ (with sup norm). Prove that $X$ has to be finite-dimensional.
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Basis of eigenfunctions in Banach space

I have a question about proving the existence of a basis of eigenfunctions. Assume we have a compact operator $L:\mathcal{B}\rightarrow\mathcal{B}$, where $\mathcal{B}$ is a Banach space of analytic ...
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Definition of continuous spectrum of a bounded operator

Let $T$ be a bounded operator acting on a Banach space $X$. The point spectrum $\sigma_p(T)$ is of $T$ is defined to be $$\sigma_p(T):=\{\lambda\in\mathbb C~|~T-\lambda\text{ has nonempty kernel}\}$$ ...
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Dual operator of an isometry

If $X,Y$ are Banach spaces and $\phi:X\to Y$ is an isometry, show that $\phi^*$ is surjective. I can use the equality $^\perp(ran \phi^*) = \ker\phi=\{0\}$, and also use the fact that $ran \phi^*$ ...
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The image of Banach space under its embedding provided by the Banach-Mazur theorem

It is a very nice argument of Banach and Mazur which they use to show that every Banach space $X$ is isometric to a subspace of the space $C(B_{X^*})$, where $B_{X^*}$ is the unit ball of the dual ...
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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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1answer
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$A^{\ast}$ compact $\Rightarrow A$ compact

I read that if $A:E\to E$ is a bounded linear operator where $E$ is a Banach space and $A^{\ast}$ is a compact operator, then $A$ is a compact operator. I know that the converse is true (th. 4 here), ...
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Compactness of the image of $S^{\ast}$ throw compact operator

I read, on an Italian language version of Kolmogorov-Fomin's Introductory Real Analysis, that, for any Banach space $E$, the unit closed sphere $S^{\ast}$ of $E^{\ast}$ is compact in the $\ast$-weak ...
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Non-separable Hilbert spaces in duals

A topological space $X$ satisfies the countable chain condition if every family of pairwise disjoint open sets in $X$ is countable. I am looking for a reference to the following fact: Suppose that ...
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Containment of $c_0$ or $\ell_p$

Suppose that $(x_n)$ is a sequence of unit vectors in a Banach space $X$ such that $$\mbox{dist}(x_m, S_{X_n})=1$$ for all $m > n$. Here $S_{X_n}$ stands for the unit sphere $\mbox{span}\{x_1, ...
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Bounded resolvent

I am finding a somewhat obscure wording on my functional analysis text. $A$ is a linear operator on a complex Banach space $E$, which I'm almost sure to mean that $A$ is $E\to E$. Then I read that if ...
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Can we equip smooth functions ($C^{\infty}[0,1]$) to some complete norm? [duplicate]

Let $C^{\infty}[0,1]$ be the vector space of all real function $f:[0,1]\rightarrow \Bbb R$ s.t for each $n \in \Bbb N$, $f^{(n)}$ exist. Is there a norm $\|.\|$ on $C^{\infty}[0,1]$ such that ...
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Banach Algebras: Continuity of Inversion?

Context: This question is related to this thread: Spaces of Functions Given a topological space $X$ and a Banach algebra with unit $B$. Consider a continuous map $F:X\to B$ that is invertible ...
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show that every compact operator on Banach space is a norm-limit of finite rank operators (in a particular way, under the given hypotheses)

Let $X$ be a Banach space and suppose there is a net $\{F_i\}$ of finite-rank operators on $X$ such that $(a)$ $\sup_i||F_i||<\infty$ ; $(b)$ $||F_ix-x||\to 0$ for all $x$ in $X$. Show that if $A$ ...
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Reference request for the proof of the Brodskii–Milman fixed point theroem for isometries

Can any one help me to access the paper M.S Brodskii and D.P Milman, On the center of a convex set, Dokl. Akad. Nauk SSSR 59 (1948) 837–840 in Russian? or to prove the theorem If $K$ is a ...
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An example of a non-compact operator which is equal to (norm) limit of compact operators

For a normed linear space $X$ and a Banach space $Y$, the set of all compact operator from $X$ to $Y$, which is denoted by $K(X,Y)$, is normed closed in $B(X,Y)$. Is there a counterexample which ...
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Existence of a Banach space with algebraic dimension $\alpha \neq \aleph_0$

For any given cardinal number $\alpha \neq \aleph_0$, is there a Banach space $X$ with algebraic dimension $\alpha$?
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Maximal chain in the collection of all invariant subspaces for compact operator $K$

Let $X$ be a Banach space over ${\Bbb C}$, and $K\in K(X)$ ($K(X) = $ compact operators space). Show that if ${\cal L}$ is a maximal chain in the $Lat K$ ($Lat K = $ the collection of all invariant ...
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Operators on non-separable Banach spaces have non-trivial invariant subspaces

Show that if $T\in B(X)$ and $X$ is not separable, then $T$ has a nontrivial invariant subspace. I know that $\ker (T)$ and $\operatorname{ran}(T)$ are invariant $T$-subspace. So if $\ker T\neq ...
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finding the algebraic dimension of $\ell^p$ spaces

I want to know "how we can find the algebraic dimension(the cardinal number of the Hamel basis) for $\ell^p$ spaces." What can we say about $\ell^p(I)$, where $I$ is an infinite set?\ Moreover, for ...
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The space $ \left( \sum \ell_p^n \right)_2$ is reflexive.

Let $\ell_p^n:= (\mathbb R^n, \|\cdot\|_p)$. I want to show that the space $$ \left(\sum_{n=1}^\infty\ell_p^n \right)_2 := \left(\left\{ (x_n)_{n \in \mathbb N} : x_n \in \ell_p^n ...
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Density of a closed and convex subset of a Banach space.

There's this homework problem that I've been battling on-and-off for about a week now. I've managed to reduce it to the following lemma. Lemma. Let $X$ denote a Banach space, and consider closed ...
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Invariant subspaces and their complements

The following is Exercise 6.4.2 of Conway's Functional Analysis: Suppose $T\in B(X)$ where $X$ is a Banach space. Prove that $M$ is invariant under $T$ if and only if $M^{\perp}$ is invariant under ...
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Compact/open subsets of vector space

Let $K\subset U$ be compact (resp. open) subsets of a normed vector space. Must there exist an open $V$ such that $K+V\subset U$?
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Differentiation in Besov-Zigmund spaces

This is my second question in a short time on Besov spaces. I apologize. I am having a rough time with them and I really need to understand this spaces quickly. The besov spaces ...
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Compactness of an operator on $c_0$ in terms of its infinite matrix representation

Let $A\in {\cal B}(c_0)$ (${\cal B}(c_0)$ is linear bounded operators on $c_0$) and for $n\geq 1$, define $e_n \in c_0$ by $e_n(m)=\delta_{nm}$. Put $\alpha_{nm}=(Ae_n)(m)$ for $n,m\geq 1$. we have $M ...
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Composition operator of a non-analytic function

I recently came across a problem that can be reformulated in simple terms involving the composition operator $$ C_g\colon X \to X, \quad f \mapsto f \circ g $$ for functions $$ f,g\colon \mathbb{C} ...
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Proving $u$ is compact whenever $u^\ast$ is

Let $X,Y$ be Banach spaces and let $u: X \to Y$ be a linear operator. Let $u^\ast: Y^\ast \to X^\ast$ denote its transpose and assume that $u^\ast$ is compact. I am trying to prove that $u$ is ...
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Is $C^{\infty}([0,1])$ a Banach space?

I have read that the answer is no, but I am unable to prove it. Give $C^{\infty}([0,1])$ the metric $$d(f,g) = \sum_{j=0}^{\infty} 2^{-j} \frac{||(f-g)^{(j)}||}{1 + ||(f-g)^{(j)}||}$$ associated to ...