# Tagged Questions

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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### Space of $f(0)=f(1)$: Is Hilbert space?

Let $S$ be space consisting of collection of square integrable continuous functions $f:[0,1]\rightarrow\mathbb{R}$ with the constraint $f(0)=f(1)$. So $S$ is an inner product space with the inner ...
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### $Im(K)\subset Y$ closed, infinite-dimensional: $K:X\to Y$ is not a compact operator

Proposition: If $K:X\to Y$ is a bounded linear operator between two Banach spaces $X$ and $Y$ such that $\operatorname{Im}(K)\subset Y$ is an infinite-dimensional closed subspace, then $K$ is NOT ...
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### How to prove $\limsup\limits_{n\to\infty}\rho_k(x_n+x)=\limsup\limits_{n\to\infty}\rho_k(x_n)+\rho(x)?$ on $\ell_1$

Let $p(.)$ be an equivalent norm to the usual norm on $\ell_1$ such that $$\limsup\limits_{n\to\infty} p(x_n+x)=\limsup\limits_{n\to\infty}p(x_n)+p(x)$$ for every $w^*-$null sequence $(x_n)$ and for ...
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### Proving that $c$ is a Banach space.

I want to prove that $$c = \{ (x_n)_{n\geq 0} \mid x_n \in \Bbb C \text{ and the sequence converges} \}$$ with the norm $$\left\|(x_n)_{n\geq 0}\right\|_{\infty} =\sup_{n\geq 0} |x_n|$$ is a Banach ...
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### About the Banach algebra $\ell^{\infty} (K( \ell^{1}))$

If $P_{n}: \ell^{1} \rightarrow \ell^{1}$ is the projection onto the first n coordinates, then it's well known that $P_{n} K(\ell^{1}) P_{n}$ is isomorphic to $B(\ell^{1}_{n})$, the Banach space of ...
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### Determining whether equality $\|T v\| = \|T\| \cdot \|v\|$ is possible

As an exercise, I'm supposed to determine whether for the operators $A_\lambda:g(t)\mapsto \sqrt\lambda g(\lambda t)$ on $C[0,1],\lambda\in (0,1)$ and $T_f:g\mapsto g(t)f(t)$ on $L^2$ it is possible ...
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### Implicit function theorem for Banach spaces

I was wondering if someone could give a bit of broad advice regarding working with Implicit Function Theorem (IFT) and, I guess, the Catastrophe theory. This is something completely new to me. ...
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### What's the difference between the topology defined by a seminorm and the topology defined by the norm it induces?

I was just wondering whether there's some big difference between the topology generated by a seminorm and the norm it induces. For instance, Suppose $X$ is normed and $A$ is a subspace. $X/A$ is semi-...
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### Is a reflexive space necessarily an L-embedded space?

I am reading some paper about L-embedded space. For the definition of L-embedded space, see http://www.sciencedirect.com/science/article/pii/S0022247X02001075. Let $Y$ be a Banach space and $P$ a ...
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### Existence of a function that fulfills an equation

I am just revising for my exams and came across this question: Show that in the Banach-space of functions that are continuous in the interval $[-1,1]$, together with the supremum-norm, there is ...
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### Do null sequences in Banach space have summable subsequences?

One of the very nice features of null scalar sequences is the fact that they admit summable subsequences. Is the same true in Banach spaces? That is, if $(x_n)_{n=1}^\infty$ is a sequence in a Banach ...
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### Tensor product of algebras of operators on $SQ_p$ spaces

I am aware that there is a canonical tensor product for $L_p$ spaces that allows one to talk about tensor products of algebras of bounded linear operators on $L_p$ spaces. Is there an analogous notion ...
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### Fixed points and banach spaces [closed]

Let $B$ be a closed ball centered at $0$ in a Banach space $E$ and $F:B\to E$ be a contractive map such that $F(x)=-F(-x)$ for every $x\in\partial B$. Show that $F$ has a fixed point. Any ideas?
Suppose we work in a separable, complete metric space $X$. Let $Z$ be an uncountable subset of $X$, must there exist $x_0\in Z$ and a sequence $(x_n)_{n=1}^\infty$ of elements in $Z$ different from $... 1answer 34 views ### Summation functional on a Hamel basis Let$X$be an infinite-dimensional Banach space. Is it possible to choose a Hamel basis$B$of$X$such that the linear functional defined by$f(b)=1$($b\in B$) was continuous? 1answer 21 views ### Strongly measurable implies Borel measurable in separable space Let$(M,\mu)$be a measure space,$X$be a Banach space,$f: M \to X$be a function.$f$is said to be strongly measurable if there is a sequence of simple functions$\{f_n\}\to f$pointwisely a.e..$...
I've been reading the GTM text Topics in Banach Space Theory by Albaic and Kalton. In the appendix, it states the following partial converse to the Banach-Steinhaus theorem: Let $\{S_n\}$ be a ...