# 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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### Does the open mapping theorem imply the Baire category theorem?

A nice observation by C.E. Blair1, 2, 3 shows that the Baire category theorem for complete metric spaces is equivalent to the axiom of (countable) dependent choice. On the other hand, the three ...
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### How slow/fast can $L^p$ norm grow?

This is actually an exercise in Rudin's Real and Complex Analysis, $L^p$ spaces chapter. Could anyone help me out? Thanks in advance. Motivation: It's well known that if we have a function $f$ which ...
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### Is it possible to characterize completeness of a normed vector space by convergence of Neumann series?

If $X$ is a normed vector space and if for each bounded operator $T \in B(X)$ with $\| T\| < 1$, the operator ${\rm id} - T$ is boundedly invertible, does it follow that $X$ is complete? Context: ...
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### Where does the theory of Banach space-valued holomorphic functions differ from the classical treatment?

For a Banach space $V$ over $\mathbb{C}$ and $U \subset \mathbb{C}$ open, one can easily check that the notions of holomorphy hold for maps $f: U \rightarrow V$ just as in the classical sense. Indeed, ...
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### Example of a closed subspace of a Banach space which is not complemented?

In this post, all vector spaces are assumed to be real or complex. Let $(X, ||\cdot||)$ be a Banach space, $Y \subset X$ a closed subspace. $Y$ is called $\underline{\mathrm{complemented}}$, if there ...
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### Motivation for abstract harmonic analysis

I am reading Folland's A Course in Abstract Harmonic Analysis and find this book extremely exciting. However it seems Folland does not give many examples to illustrate the motivation behind much of ...
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### Norms on C[0, 1] inducing the same topology as the sup norm

This is an old homework problem of mine that I was never able to solve. The solution may or may not involve the Baire category theorem, which I am terrible at applying. Let $C[0, 1]$ denote the ...
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### Reinventing The Wheel - Part 2: The Lebesgue Integral

Disclaimer After struggling for some time to find an appropriate definition for the notion of integration I came across another attempt for which I would need your help deciding to what extend this ...
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### Different versions of Riesz Theorems

In Wikipedia, there are three versions of Riesz theorems: 1 The Hilbert space representation theorem for the (continuous) dual space of a Hilbert space; 2 The representation theorem for ...
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I have a linear operator $A\in\mathcal{L}(X,Y)$ where $X$ and $Y$ are some Banach spaces (or Hilbert spaces would also do, if that simplifies the answer.). The operator norm of $A$ is given by $$\|A\... 1answer 1k views ### Are the coordinate functions of a Hamel basis for an infinite dimensional Banach space discontinuous? The question is in the title really, but I suppose I could at least fix some notation here. Let X be an infinite dimensional Banach space - over the reals for the sake of concreteness. Use choice ... 0answers 480 views ### defining a topology by its compact sets The goal. Let X be a set endowed with Hausdorff topologies \tau_w and \tau_n, such that \tau_w\subseteq\tau_n. Let \mathscr{C} denote a family of subsets A\subseteq X, which satisfies ... 4answers 1k views ### Banach spaces over fields other than \mathbb{C}? Sorry, this is a rather vague question. I was just wondering if there is any kind of theory about normed (if possible Banach) spaces over fields other than the real or complex numbers. I'm guessing ... 1answer 3k views ### How to deduce open mapping theorem from closed graph theorem? These two theorems are equivalent but I can not figure out how to deduce the open mapping from the closed graph. Can anyone give a hint or some reference? 1answer 11k views ### Prove that C^1([a,b]) with the C^1- norm is a Banach Space Consider the space of continuously differentiable functions,$$C^1([a,b]) = \{f:[a,b]\rightarrow \mathbb{R}|f,f' \text{are continuous}\}$$with the C^1-norm$$||f|| = \sup_{a\leq x\leq b}|f(x)|+\...
Let $B$ be a (complex) Banach space. A function $f : \mathbb{C} \to B$ is holomorphic if $\lim_{w \to z} \frac{f(w) - f(z)}{w - z}$ exists for all $z$, just as in the ordinary case where \$B = \mathbb{...