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.
6,740 questions
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When is the Frechet derivative invertible for a continuously differentiable homeomorphism between Banach spaces
Let $X, Y$ be Banach and $U\subset X, V\subset Y$ open. Suppose $f: U\to V$ is a homeomorphism and $f\in C^1(U)$. If for $x_0\in U$, $f'(x_0)\neq 0$ and $f'(x_0)$ is bounded below, is it then ...
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$\lambda^n \in \rho(A^n)\Rightarrow \lambda \in \rho(A)$, where $\rho(A)$ is resolvent set of $A$.
Let $X$ be a complex Banach space and $L(X)$ the set of bounded linear operators over $X$.
For a bounded linear operator $A: X \rightarrow X$, we define
$$
\rho(A)=\{\lambda \in \mathbb C: (\lambda I ...
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Proposition 3.1.5 in Topics in Banach Space Theory (Characterisation of Unconditional Bases)
I am confused about Proposition 3.1.5 in the book Topics in Banach Space Theory by Albiac and Kalton.
We let $(u_n)$ be a Schauder basis for a Banach space $X$. Let the associated coordinate ...
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Why must a homomorphism between Banach spaces be bounded? [closed]
Let $V$ and $W$ be vector spaces. Then a homomorphism $f: V \rightarrow W$ is any map that preserves the vector space structure on $V$ so $f(av_1 + bv_2) = af(v_1) + bf(v_2)$. If we take $V$ and $W$ ...
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Characterization of weakly unconditionally Cauchy series
I am reading from Bases in Banach spaces I by I. Singer. The below defintions and results are from this book.
Let us first recall that a series $\displaystyle \sum_{i=1}^\infty x_i$ in a Banach space ...
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Exercise 4.34(a) from An Introduction to Banach Space Theory by Megginson
This is Exercise 4.34(a) from An Introduction to Banach Space Theory by Robert E. Megginson.
Give an example of a sequence $(x_n)$ in a Banach space $X$ such that $(x_n)$ is not basic even though $\...
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Meaning of stronger hypotheses
This is from An Introduction to Banach Space Theory, by Robert E. Megginson.
Theorem 4.3.6
Suppose that $(x_n)$ is a sequence in a Banach space. Then $(x_n)$ is a basic sequence equivalent to the ...
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When are homeomorphic topological vector spaces isomorphic?
I'm now interested in when homeomorphic implies isomorphic (i.e., linearly isometic) for certain classes of topological vector spaces especially over $\mathbb{C}$.
I knew that homeomorphic Hilbert ...
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Help me by providing a few references on the study of isometric groups between two Banach spaces
I am interested in studying isometric groups between Banach spaces. Let $X$ and $Y$ be Banach spaces. Denote by $\operatorname{Iso}(X, Y)$ the collection of all surjective isometries from $X$ onto $Y$....
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Existence of Schauder basis for the Banach space of bounded linear operators on $L^2$
This question stems from my study of the functional data analysis (FDA) field of statistics. In FDA one deals with infinite-dimensional objects, such as functions or surfaces. The space $L^2\equiv L^2(...
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T/F: $X, Y$ Banach spaces. $T\in L(X, Y)$ is a compact norm-attaining operator $\iff T^*\in L(Y^*, X^*)$ is also a compact norm-attaining operator.
Let $X$ and $Y$ be Banach spaces. Denote $L(X, Y)$ as the space of bounded linear operators from $X$ to $Y$. Then is the following result true:
$$ T\in L(X, Y) ~\text{is compact and norm-attaining} \...
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Weak Convergence in completions of a Pre-Hilbert space with respect to different norms
Let $K$ be a pre-Hilbert space equipped with two norms, $\tau_\ell$ and $\tau_h$, satisfying
$$
\tau_\ell(\xi) \leq \tau_h(\xi) \quad \text{for all } \xi \in K.
$$
Let $(L, \tau_\ell)$ and $(H, \tau_h)...
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Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be a linear map. Is every bounded subset of $\text{Im}(f)$ the image of some bounded set.
Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be a linear map. Let $B\subseteq \text{Im}(f)$ be a bounded set. Is there always a bounded set $A\subseteq \mathbb{R}^n$ satisfying that $f(A)=B$?
If n=1 ...
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If $\lambda \in \bigcap_{\delta > 0} ~\overline{\left\{ f(T) : f\in B_{L(X, Y)^*}, |f(G)| > 1 - \delta \right\}}$, $T\in L(X, Y)$ then $\lambda =$?
Let $X, Y$ be Banach spaces. Let $L(X, Y)$ be the collection of bounded linear operator from a $X$ to $Y$. We denote $B_X$ as the closed unit disk of $X$. Suppose that $G\in L(X, Y)$ such that $\|G\|=...
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Two definitions of summable series in a Hilbert space
I am trying to prove a comment made on page 18 in Halmos's Introduction to Hilbert Spaces and The Theory of Spectral Multiplicity. For below, assume $X$ is an arbitrary Hilbert space. I am rephrasing ...
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