Questions tagged [functional-analysis]

Functional analysis, 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 and other topics. For basic questions about functions use more suitable tags like (functions), (functional-equations) or (elementary-set-theory).

32,040 questions
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Is the norm $||x||:= \sum\limits_{i=1}^{\infty} 2^{-i}|x_i|$ equivalent to usual norm in $l_2$?

Is the norm $$||x||:= \sum_{i=1}^{\infty} 2^{-i}|x_i|$$ equivalent to usual norm in $l_2$? I have already shown that it is a norm, I expect that it is equivalent, but I can not prove it.
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Distance to set

Let $S$ be a non empty set in an inner product space $E$ Show that if $x\in E,z\in E$ and $Re<x-z, y-z> \le0$ for each $y\in S$ then $d(x,S)=||x-z||$ I would like a clue on how to approach ...
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structure of subrepresentations of (infinite) sums of irr. representations

Let $G$ be a (locally compact) group and $( \pi_1 , V_{\pi_1} ) , ( \pi_2 , V_{\pi_2} ) , \ldots$ irreducible, unitary representations. I think, I can show, that for the (finite) direct sum ...
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If $A \in \mathcal{L}(H)$ and $\langle A(u),u \rangle \geq \langle u, u \rangle$, then $A$ is invertible.

Exercise : Let $H$ be a Hilbert space and $A \in \mathcal{L}(H)$ such that : $$\langle A(u),u \rangle \geq \langle u, u \rangle \; \forall u \in H$$ Show that $A$ is invertible. Attempt/...
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Are the bounded Lipschitz functions dense in $L^1(\mu)$?

I am currently reading a paper (L. Ambrosio and B. Kirchheim. Currents in metric spaces) and I stumbled uppon a fact which I don't know how to prove. I have the following setting: Let $X$ be a ...
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Spectrum of an operator defined by spectral integral

First of all I want to thank you for the help you provide on this website! Whenever I had a hard time understanding things in math I visited this website and (nearly) allways found a hint or a ...
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Convergence in $W^{1,\infty}$ implying convergence in $W^{-1,\infty}$.

I'm looking for references or proofs of the following facts: Weak star convergence in $L^\infty$ implies strong convergence in $W^{-1,\infty}_{\text{loc}}$, and Strong convergence in $L^\infty$ + ...
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Convergence of Indicator function in weak*-topology

Let $\omega_n$ and $\omega$ be measurable subsets of $[0,1]$. Also let indicator function $\chi_{\omega_n}$ converge to $\chi_{\omega}$ in weak*-topology on $L^\infty(0,1)$, that is \int_0^1f(x)(\...
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Weak-* continuous functionals with bounded level sets

Let $X$ be an infinite-dimensional Banach space, $X^*$ its topological dual and $f:X^*\to\mathbb{R}$ some weak-* continuous functional (not necessarily linear). Is it possible for $f^{-1}(a)$ to ...
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Riemann's Lemma as a corollary of Bessel's Inequality

I am learning about Fourier Series with Carother's Real Analysis. I'm currently stuck on the bolded statement below made by the author. First I will give a little context. We have just proved the ...
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Bijection+Closability?

I was wondering whether a closable operator $A$ with domain $D(A)\subset H$ ($H$ is a Hilbert space) which is also bijective has an everywhere defined bounded inverse? Thanks in advance. Math.