# 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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### Right Hilbert space for a duality pairing

Let $B$ be a Banach space. Let $B^*$ be its dual, and let $K\subseteq B^*$ be some linear dense subspace. Denote the duality pairing between $B$ and $B^*$ via $\langle\cdot ,\cdot \rangle$. Suppose we ...
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### Compactness when mapping into a higher $L^p$ space and then back

Question: Let $q>p \ge 1$ and let $T:L^p[0,1] \to L^q[0,1]$ be a bounded linear operator. Let $i: L^q[0,1] \to L^p[0,1]$ be the inclusion map (which is bounded). Is the composition $i\circ T$ ...
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### Proving that the set of continuous linear maps (From a Banach space $X$ to another Banach space $Y$) is an open subset of $L(X,Y)$

I am a university undergraduate student, and I am reading up on the German Functional Analysis textbook "Introduction to Functional Analysis" by Friedrich Hirzebruch/Winfried Scharlau. My question ...
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### Convergence of a series with general index set

I saw that it is possible to define the convergence of a sequence indicized by a general set (with an order relation) using NETS. My question is if it is possible to define in this way the notion of ...
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### $(C^1[0,1], || . ||_\infty)$ is not Banach Space.

Recently, I am studying Banach Space. I know that $(C[0,1], || . ||_\infty)$ is a Banach Space and closed subspace of Banach Space is Banach. This follows from the result that closed subset of ...
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### If $T:L^p[0,1] \to L^p[0,1]$ bounded for $1 < p < \infty$ with continuous image, then it's compact

Is the following statement true? Let $T:L^p[0,1] \to L^p[0,1]$ be a bounded operator for $1 < p < \infty$ and suppose that $\operatorname{Im}(T) \subset C[0,1]$ consists of continuous functions....
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### A question regarding Radon-Riesz property

I want to show that if a normed linear space has Radon-Riesz property, then it has the semi Radon-Riesz property. We know that a normed linear space is said to have Radon-Riesz property if for every ...
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### Quotient space of a a normed linear space with schur's property need have have that property.

I want to show that if a normed linear space has Schur's property (every weakly convergent sequence converges), then it is not guaranteed that every quotient space $X/Y$ will have that property, where ...
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### Is the set of piecewise $C^1([a,b])$ functions complete with $C^1$ norm?

Here's the $C^1$ norm : $|| f || = \sup | f | + \sup | f '|$ where the supremum is taken on $[a, b]$. Please, justify your answer (proofs or counterexamples are needed).
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### Is there a norm making $C([0,1])$ into a Hilbert space?

The space $C([0,1])$ of continuous functions on $[0,1]$ is an inner product space under the $L^2$-norm, but not complete. Equipped instead with the $L^\infty$-norm, it becomes complete but the norm is ...
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### Books about fixed-point theory

I am looking for book recommendations about Fixed-Point Theory. I found this post: Book Recommendation for Iterated Functions? recommending a book by Shashkin. However, I cannot find who Shashkin is,...
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### From $\sup_{n\in \mathbb{N}} \left|\int_0^\Lambda e^{nx} f(x) dx \right| < \infty$ to $f\equiv 0$

Given $f\in C[0,\Lambda]$ satisfying $$\sup_{n\in \mathbb{N}} \left|\int_0^\Lambda e^{nx} f(x) dx \right| < \infty$$ Prove that $f\equiv 0$ $\,\forall x\in[0,\Lambda]$ I found a weaker ...
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### Complement a finite dimensional subspace in a Banach space

Given a Banach space $(X,\|,\|)$, and a finite dimensional subspace $F \subset X$, is it always possible to choose a closed linear complement to $F$. Explicitly, I mean to say, will there always exist ...
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### Characterization of Reflexive Banach Space.

Prove that a real Banach Space $X$ is reflexive if and only if each pair of disjoint closed, convex subsets of $X$, one of which is bounded, can be strictly separated by a hyperplane. The theorem is ...
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### Counterexample for “continuous image of closed and bounded is closed and bounded” (in normed spaces).

It's well known that: If $X$ is a finite-dimensional normed space, $C$ is a closed and bounded subset of $X$ and $f:C\subset X\to X$ is continuous, then $f(C)$ is closed and bounded. If $X$ is any ...
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### Boundedness of a Net in Second Dual Transfering to Original Space

Let $X$ be a Banach space. Let $J : X \to X^{**}$ denote the natural embedding of $X$ into $X^{**}$. Suppose that there exists a bounded net $(T_{\alpha})_{\alpha \in I} \subset X^{**}$ ...
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### Example of a topological space $X$ such that $C_0 (X)$ is not a $C^*$-sub-algebra of $C^b (X)$

Let $X$ be an arbitrary topological space. If $X$ is locally compact and Hausdorff, then $C_0 (X)$ (space of continuous functions vanishing at infinity) is a $C^*$-sub-algebra of $C^b (X)$ (space of ...
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### Closed ideal in $L^{1}(G)$

Let $G$ be locally compact group prove that $$L_{0}^{1}(G)=\left\{f\in L^{1}(G): \int_G f(g) dm(g)=0 \right\}$$ is a closed ideal in $L^{1}(G)$ with codimension one I am grateful for any ...
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### What does it mean for an operator to be diagonal with respect to an orthonormal basis?

Suppose that you have a seperable Hilbert space, $H$, an orthonormal basis of $H$, $(e_n)$, and the set $D=\{T\in B(H):T\,\text{is diagonal with respect to the basis}\,(e_n)\}$. Problem: What, ...
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### Why are these two characterizations of compact operators equivalent?

If $T$ is a bounded linear operator on a Hilbert space $H$, then I have heard that the following two things are true: $T$ is compact if and only if $T(C_1)$ is compact where $C_1$ is the closed unit ...