# Tagged Questions

Functional analysis is the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces, as well as measure, ...

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### Open, convex set of TVS

I'm studying LCS using Conway's book. And I had a question about a proof of Proposition 3.2 in chapter 4. The author said, the proof of this proposition is similar to that of proposition 1.14 (If V ...
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### What's so special about $p=2$ for the $L^p$ spaces?

The Banach space dual of $L^p$ is $L^q$, where $q=\frac{p}{p-1}$, but I don't really understand the motivation behind this. In particular, I find it kind of surprising that the only $L^p$ space whose ...
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### Is Riesz measure an extension of product measure?

Suppose $X$ and $Y$ are compact Hausdorff spaces and $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ are finite regular Borel measure spaces. (By regular I mean that every measurable set can be ...
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### Does (L) sets in a dual Banach space X* are weak* precompact? weak* sequentially precompact?

Let $X$ be a Banach space. A subset $B$ of the dual $X$ is said to be $(L)$ set if any weakly null sequence $(x_n)\in X$ converges uniformly to zero on $B$. It is well Known in the theory that ...
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### Equation in Hilbert space

Solving the following exercise of a list I have: "$H$ is a complex Hilbert space admitting an orthonormal basis $\{e_n\}, n\in \mathbb{N}$ ; $\{\lambda_n\}\subset \mathbb{C}\setminus \{0\}$ is a ...
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### Is the normalized duality mapping symmetric?

In the area of functional analysis, nonlinear operator theory, the normalized duality mapping on a Banach space $X$ is a set valued map from $J:X\rightarrow 2^{X^*}$ given by \begin{align} J(x)=\{j(...
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### $\sin$ and $\cos$ are the basis of what space?

When learning Fourier expansions, we learn that $\{\sin(mx), \cos(mx)\}_{m \in \Bbb N}$ is an orthonormal basis for our space and thus we can expand functions in it. My question is what space is this ...
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### Non-negative functions are a closed subset of $C_b(K)$
Consider the Banach algebra $A=C_{b}(K)$ of all complex-valued bounded continuous functions on a completely regular Hausdorff space $K$ with the supremum norm, and let $C$ be the set \$C:=\{g \in A: g(...