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Mathematics Stack Exchange

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.

A Banach space, named after Stefan Banach (1892–1945) is a complete normed vector space: a (real or complex) vector space equipped with a norm such that every Cauchy sequence converges. For instance, $\mathbb{R}^n$ and $\mathbb{C}^n$, equipped with the usual norm (or, for that matter, any norm) is a Banach space. Another example is the space $\ell^1$ of all absolutely convergent series of real or complex numbers, equipped with the norm $\left\|\sum_{n=0}^\infty x_n\right\|=\sum_{n=0}^\infty|x_n|$.

6435 questions
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Hahn-Banach extension of function on $\mathbb{R}^2$ with $\|\;.\,\|_1$-norm

Consider $\mathbb{R}^2$ with $\|\;.\,\|_1$-norm and $M=\{(x,0) \mid x \in \mathbb{R}\}$. Define $g:M \to \mathbb{R}$ by $g(x,y)=x$. Then a Hahn-Banach extension $f$ of $g$ is given by a) $f(x,y)=2x$ b) $f(x,y)=x+y $ c) $f(x,y)=x-2y $ d)…
Halima.Khatun
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A singleton as domain sum of a series

Consider the series $$e_1+\frac{1}{2}e_2-\frac{1}{2}e_2+\frac{1}{4}e_3-\frac{1}{4}e_3+\frac{1}{4}e_3-\frac{1}{4}e_3+\frac{1}{8}e_4-\frac{1}{8}e_4+\cdots-\frac{1}{8}e_4+\frac{1}{16}e_5-\cdots$$ in the space $\ell_2$. In example 2.2.1 of monograph…
Ana María CG
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Proof, that a set is not convex

I try to solve the problem 106 of the scottich book. I know the set of all rearranged sums is convex. Let $f_{j,k}$ the indicator function of the interval $(\frac{j}{2^k},\frac{j+1}{2^k}$). k = 0,1,2,$\cdots$ and j = 0..$2^k-1$. I have proofed,…
TheK
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Weak$^*$-convergence of vector-valued measures implies weak$^*$-convergence in $X^*$?

Let $K$ be a compact Hausdorff space and $X$ be a Banach space. By the Riesz-Singer representation theorem, we know that there exists a linear isometry from $C(K,X)^*$ onto $rcabv(K,X^*)$, the Banach space of all regular, countably additive, Borel…
Vinícius Morelli
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countable intersection of closed convex bounded subsets reflexive banach space is non empty.

If $X$ is a reflexive Banach space and $(C_n), n \in \mathbb{N}$ is a sequence of closed convex bounded sets with $C_{n+1}$ contained in $C_n$ for all $n \in \mathbb{N}$. How does one show that the countable intersection of $C_n$ for $n \in…
nada
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Determining if linear operator on space of polynomials is bounded

I have $p$ a polynomial given by $p(x) = a_0 + a_1 x + a_2 x^2 ... a_n x^n$ and a linear operator $T$ defined by $T(p)(x) = a_0 + a_1 x^2 + a_2 x^4 + ... + x^{2n}$. The norm on the space is given by $||p|| = \sup \{|p(t)| :t\in[0,1] \}$. I'm trying…
Wooster
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Prove: $M +N$ is closed in $B$ iff $∥m∥_B +∥n∥_B ≤ c∥m+n∥_B$, for all $m ∈ M$ and $n ∈ N$.

Let $B$ be a Banach space and $M,N$ closed subspaces of $B$ such that $M ∩N = \{0\}$. Prove: $M +N$ is closed in $B$ iff $∥m∥_B +∥n∥_B ≤ c∥m+n∥_B$, for all $m ∈ M$ and $n ∈ N$. My Work: If $Y$ is a subspace of $B$ then $Y$ is complete iff $Y$…
Extremal
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Showing bound on operator $||Tx|| \ge \frac{||x||}{||(T')^{-1}||}$

T is an bounded linear operator on a Banach space, X, and I'm given that it's adjoint $T'$ is invertible, I'm trying to show: $||Tx|| \ge \frac{||x||}{||(T')^{-1}||}$. I feel like this should be simple but I'm finding it very frustrating! My ideas…
Wooster
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