# 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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### Dual space of the space of finite measures

Since I am reading some stuff about weak convergence of probability measures, I started to wonder what is the dual space of the space consisting of all the finite (signed) measures (which is well ...
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### Liouville's theorem for Banach spaces without the Hahn-Banach theorem?

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{...