# 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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### Spectrum of the Laplacian for free space versus spectrum of Laplacian with boundary conditions?

In this question the spectrum of the Laplacian in free space is defined as $]-\infty, 0]$. So it seems the spectrum can be determined without regard to boundary conditions? If instead we consider the ...
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### If $\mathcal H$ is the closure of the set $D$ of divergence-free smooth functions in $L^2$, then $H_0^1∩\mathcal H$ is the closure of $D$ in $H_0^1$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D:=C_c^\infty(\Omega,\mathbb R^d)$ and $$\mathfrak D:=\left\{u\in\mathcal D:\nabla\cdot u=0\right\}$$ $H:=H_0^1(\Omega,\mathbb R^d)$...
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### $\ker ST=\ker T$

Let $S$ and $T$ be linear maps between vector spaces such that the composition $ST$ makes sense. Clearly, $\ker ST\supseteq \ker T$. The two instances that come to my mind for having an equality in ...
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### Continuous semi-norms on subspace

Suppose $X$ is a locally convex topological vector space, let $P$ be the set of all continuous semi-norms on $X$. Suppose $M$ is a subspace of $X$, denote $P|_M$ as the set of semi-norms in $P$ ...
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### Complex interpolation between $H^1$ and $L^1$

We have the complex interpolation $[H^1,L^p]_\theta=L^q$ for $1<p<\infty$ where $H^1$ is the Hardy space and $\frac{1}{q}=1-\theta+\frac{\theta}{p}$. This raises the obvious question as to ...
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### Tensor products $L_1(\mu)\widetilde{\otimes}_{\pi} L_1(\nu)$ and $L_1(\mu)\widetilde{\otimes}_{\varepsilon} L_1(\nu)$

Can anybody enlighten me, where the tensor products of the spaces of summable functions $L_1(\mu)\widetilde{\otimes}_{\pi} L_1(\nu)$ and $L_1(\mu)\widetilde{\otimes}_{\varepsilon} L_1(\nu)$ are ...
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### Proof validation: complete set, change of variable

Let $\phi(x) \in \mathcal{C}^1([0,1])$ be a real valued function such that: $$\begin{cases} \phi'(x) > 0 & \forall x \in [0,1] \\ \phi(0) = 0 \\ \phi(1) = 1. \end{cases}$$ I'm asked to prove ...
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### Topology for Hardy spaces

Let $\Omega\subset \mathbb{C}$ be an open set (of the complex plane) and let $\mathcal{H}(\Omega)$ be the algebra of analytic functions on $\Omega$ endowed with the topology of compact convergence (...
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### A basis of $L^2$

I would like to ask you a question that is there a basis of the space $L^2(\Omega,\mathcal{F},\mathbb{P})$, where $\mathbb{P}$-probability measure?
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### Bounding $\int_0^1 f(x) dx$ under the condition $\int_0^1 f'(x)^2 dx \le 1$

Any tips on how to solve this? Problem 1.1.28 (Fa87) Let $S$ be the set of all real $C^1$ functions $f$ on $[0, 1]$ such that $f(0) = 0$ and $$\int_0^1 f'(x)^2 dx \le 1 \;.$$ Define ...
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### Elementary proof of $C^\infty_c$ is dense in $L^p (L^q)$ mixed space

As it well known, $C^\infty_0 (\mathbb{R}^n)$(the space of infinitely differentiable functions with compact support) is dense in $L^p (\mathbb{R)^n}$ Here I want to consider the same result with ...
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### Using scaling arguments to determine relationships between Sobolev spaces?

I was looking up how to find relationships between Sobolev spaces and I came across this post on MO in which the first comment talks about a scaling procedure for understanding the relationships: ...
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### What is the difference between sequence space of bounded complex sequences compared to sequence spaces of bounded and unbounded complex sequences?

Both spaces have different measures as folllows (click to see) Bounded sequence space measure and Bounded and unbounded sequence space measure
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### Finding closure of image of operator

I'm working on an old exam problem: Define for $u \in C^2([-1,1])$ the operator $L$ by $[Lu](x) = - \frac{d}{dx} \left( (1-x^2) u'(x) \right)$. Set $\Omega = \{ Lu \mid u \in C^2([-1,1]) \}$. Find the ...
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### Extension of semi-norm in locally convex topological space

Suppose $X$ is a locally convex topological space,$M$ is a subspace. Suppose $p$ is a continuous semi-norm on $M$. Is it possible to extend $p$ to be a continuous semi-norm on $X$? What if $X$ is not ...
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### Uniform closure of densely continuous functions

Consider the collection of those $\mathbb{R}$-valued functions on an interval $I\subseteq\mathbb{R}$, which have a dense set of points of continuity. I would expect this collection to be closed under ...
Prove the equivalence between: $\forall x \in B_E = \{y \in E:\|y\| \leq 1 \}$ $\exists (x_n) \subset E$ such that $\|x_n\|=1$ and $x_n \rightarrow_w x$ (weak convergence). $\exists (x_n) \subset E$ ...
Suppose that $A, B \subset B(\mathcal{H})$ are $C^*$-algebras. Assume that $\{p_n\} \subset B(\mathcal{H})$ is a monotone sequence of projections such that: $p_n \rightarrow 1$ in strong operator ...