# 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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### A question about the natural identification between $Y$ and $Y^{**}$, with $Y$ normed space. Is the following fact obvious?

Let $X$ be a reflexive normed space, $Y$ a normed space and $T: X \to Y$ a linear operator. I consider a sequence $\{x_n\}$ in X and the sequence $\{T(x_n)\}$ in $Y$. I know that there is a ...
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### Spectrum of periodic schrödinger operators

In many articles it's stated, as if it's common knowledge, that any Schrödinger operator with periodic potenial has purely absolutely continuous spectrum. I've tried to actually find a theorem ...
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### Modifying a smooth function with respect to conditions on its partial derivates

Let $\{U_i\}_{i\in I}$ be a locally finite collection of open subsets of $\mathbb{R}^n$, $K_i\subseteq U_i$ compact subsets, $\epsilon_i>0$ positive real numbers and a nonnegative natural number $k$...
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### Usefulness of Functional analysis

I heard that functional analysis can be applied to many problems in signal processing. I'm trying to explain to my engineer friend why it is useful, but I learnt it in a pure math setting. Can anyone ...
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### Uniform convergence and equicontinuity

Given a sequence of functions which is not uniformly convergent, can we deduce, that none of its subsequences is uniformly continous and therefore, by Arzela-Ascoli say that the family of function is ...
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### If the image of a linear transformation of normed spaces is finite dimensional, is the map bounded?

Let $V, W$ be normed spaces. If $T: V \rightarrow W$ is such that $T(V)$ is finite dimensional, does it follow that $T$ is bounded? Edit: This isn't a homework question, I'm just asking because I'm ...
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### Why is every $H^1$ function on the circle this way?

I want to know, specifically, why is every $H^1$ function which is defined on the circle a absolutely continuous function, with square-integrable derivative defined almost everywhere. I have no ...
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### Spectrum of simple multiplication operator on $L^2(0,1)$
I'm trying to calculate the spectrum of the linear operator $T: L^2(0,1) \to L^2(0,1)$ given by $T(f) \to tf(t)$. I've found a few facts about this operator but I'm still struggling to find the exact ...