# Questions tagged [banach-spaces]

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 a Banach Space.Prove that, $X$ is strictly convex iff every points of $S(X)$ is exposed points of $B(X)$.

Some definitions- $X$ is said to be strictly convex or rotund if for all $x,y\in S(X),\ x\ne y$ we have $\left\lVert\frac{x+y}{2}\right\rVert<1$ A point $x_0\in B(X)$ is said to be a exposed point ...
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### How to prove that the convergence of $(f_n)$ is uniform on compact sets?

I'm reading Theorem 0.9 in this lecture note. Below is my attempt where I got stuck at the end. Could you elaborate on how to finish the proof? Let $C$ be an open convex subset of a Banach space $X$....
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### $X$ is strictly convex (rotund) iff for all $x,y\in S(X)$ with $x\ne y$ we have $\lVert x+y\rVert <2$

A Banach Space $X$ is said to be strictly convex or rotund if for all $x,y\in X$ we have $\lVert x+y\rVert<\lVert x\rVert+\lVert y\rVert$ unless $x,y$ are multiple of each other. We have to prove ...
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### Relationship between Banach Space and Measurable Space

Motivation I am asking this question because in the definition of a Martingale on Wikipedia they define a random variable $f:\Omega\to S$ over a Banach space S. However typically one defines a random ...
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### Lebesgue's criterion for Riemann-integrable functions with values in a Banach space

Does anybody know a definition of Riemann-integrability for functions with values in an arbitrary (!) Banach space for which Lebesgue's criterion holds which says that a bounded function is Riemann-...
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### operator between Banach spaces and its boundeness

I am trying to understand the proof of the following: Let $T:X\to Y$ be an operator between two Banach spaces $X$ and $Y$ and $T$ is linear. Let $\mathrm{dim}\,X=n\,<\infty$. Then $T$ is bounded. ...
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### ways to calculate norm of the Bounded Linear Functional.

$\Lambda$ be a linear functional on $C([0,1])$ defined by $$\Lambda(f) = \int_0^1 xf(x)dx \;\;\; \text{ for } f \in C([0,1]).$$ and use $\|f\|_{sup} = \sup_{x \in [0,1]} |f(x)|$ for $f \in C([0,1])$...
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### Rolle's Theorem for functions defined on Banach space

In J.Dieudonne's "Treatise on analysis, vol. 1" Chapter 8.2, there's a problem which asks to prove Rolle's theorem for a function defined on a Banach Space. The problem is as follows: Let f ...
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### On the Interpretation of Volterra's theory of Integral Equations

Consider an integral equation $$u(x) - \int_a^x K(x,y) u(y) dy = f(x), \qquad x \in [a,b] \qquad (1)$$ where $u:[a,b] \to \mathbb{R}$ is an unknown function and $f$ and $K$ are known continuous ...
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### Norm convergence of series in banach space

Let $\mathfrak{X}$ be a Banach space and $\left(\rho_{n}\right)_{n \in \mathbb{N}}$ be a sequence in $\mathfrak{X}^\#$. Prove that the following assertions are equivalent: For each norm-convergent ...
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### Closed mid-point convex set and it's usefulness

$(X, \|•\|)$ be any normed space and $E\subset X$ . $(I)$ For every sequence $(x_n), (y_n) \in E$ with $x_n\to x$ and $y_n\to y$ in the space $(X,\|•\|)$ implies $\frac{x+y}{2}\in E\space \space$...
### dense subspace of $L^2$ that is disjoint to $L^p$
I wonder if it is possible to have a dense subspace $U \subseteq L^2$ that is disjoint to $L^p$ for some $p\neq 2$. I would expect that such $U$ exists, but I'm stuck finding an example.