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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).

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, including spectral theory, as well as measure, integration, probability on infinite dimensions, and also manifolds with local structure modeled by these vector spaces.

For basic questions about functions use more suitable tags like , or .

52582 questions
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About completeness of $l^{\infty}$ with respect to sup norm

Let $l^{\infty}$ be the space of all bounded sequences of real numbers $(x_n)_{n =1}^{\infty}$ with the sup norm. I have to show that $l^{\infty}$ is complete with respect to this norm. Proof: In the proof below I am confused with the sequence $x^n…
mathscrazy
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When is integration possible?

I'm not sure how to phrase this question. I'm sure I could write it in terms of operators between Frechet spaces, or something like that. Let me apologies to any analysts in advance for my lack of rigor. I'm basically interested in "how many"…
Fly by Night
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Why is Banach-Alaoglu theorem so important?

According to Lawrence Narici and Edward Beckenstein, the Alaoglu theorem is a "very important result - maybe the most important fact about the weak-* topology - [that] echos throughout functional analysis." (Source: Wikipedia) It is a well-known…
ferhenk
  • 457
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A basic question on dual space of $L^p[0,1]$

I recently started reading functional analysis on my own and have come about dual spaces and cannot get an intuitive understanding about them. This is where my intuition breaks down while understanding duals of $L^p[0,1]$ spaces for $1\le…
nadurthi
  • 185
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Show that a set is compact on $C^K[0,1]$

Show that the set of the functions $A_M:=\{f ∈ C^{k+1}([0, 1]) : \|f\|_{C^{K+1}} ≤ M\}$ is compact in $C^{k}[0,1]\ \ \forall M \geq 0$. N.B.: $$\| f\|_{C^{K+1}}=\|f\|_{C^{0}}+\|f^{(k+1)}\|_{C^{0}}$$ I started by showing that $C^{k}[0,1]$ is…
Lance
  • 401
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Closed unit ball of an infinite-dimensional Banach space is not compact

I have a few questions about the following proof taken from https://math.berkeley.edu/~sarason/Class_Webpages/solutions_202B_assign11.pdf. Prove that the closed unit ball of an infinite-dimensional Banach space is not compact. Proof. Let $B$ be an…
Teodorism
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Is Lax-Miligram theorem a generalization of Riesz representation?

Let $H$ a hilbert space with inner product $\left<.,.\right>$. We denote $\|\cdot \|$ the norm induced by the inner product. Lax-miligram tels us that there is a one-to-one correspondance between Continuous and elliptic bilinear form $a:H\times H\to…
user659895
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Does this "in between" space exist?

Let $C=C([0,1])$ be the space of continuous functions from $[0,1]$ to $\mathbb{R}$ (or $\mathbb{C}$) $B=B([0,1])$ be the space of bounded funciton from $[0,1]$ to $\mathbb{R}$ (or $\mathbb{C}$) $l_\infty$ the space of bounded sequences of real (or…
augustoperez
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$e^{iBt}e^{-iAt}$converges as operator norm

Let $A,B$ be self-adjoint operators on $H$,then we can define the strong limit $$ W=s-\lim_{t\to+\infty}e^{iBt}e^{-iAt} $$ If the limit exists, then W is called the wave operator, which is fundamental in the scattering theory. My question here is…
Tomas
  • 1,369
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Sobolev space reflexivity problem

Let $ I $ open interval of $ \mathbb{R} $ We know that the Sobolev space $ W^{1, \infty}(I) $ is not reflexive. But, is there any easy proof of this result? Thank you in advance
P. Arguello
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Vector Space with Trivial Dual

How to construct a Vector Space $E$ (non trivial) such that, the only continuous linear functional in $E$ is the function $f=0$?
Tomás
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Confusion regarding Riesz's lemma

Wikipedia (and my teacher) state Riesz's lemma as follows: Let $X$ be a normed linear space and $Y$ be a subspace in $X$. If there exists $0 < r < 1$ such that for every $x\in X$ with $||x|| =1$ , one has $d(x, Y) < r$, then $Y$ is dense in…
Bruno Stonek
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An infinite subset of the closed unit ball whose elements are more than distance 1 apart

Let $V$ be any infinite dimensional normed space. Is it always possible to find $\{x_1, x_2, \dots\}$ be a countbly infinite subset of the closed unit ball, such that $||x_i-x_j||>1$ whenever $i\neq j$? So far I have tried a few sequence spaces and…
Spook
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Proof of $f = g \in L^1_{loc}$ if $f$ and $g$ act equally on $C_c^\infty$

Let $f$ and $g$ be locally integrable, say on $R^n$ (for arbitrary open domains, just extend trivially). Suppose $\forall \phi \in C_c^\infty : \int f \phi dx = \int g \phi dx$. Let $K = supp(\phi)$. If $f,g \in L^2(K)$, we see by Hilbert space…
shuhalo
  • 7,485
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Characters and maximal ideals in $\ell^1(N)$

Determine all the maximal ideals and all the characters in $\ell^1(N)$, if we know that it is a commutative Banach algebra with component-wise multiplication and addition. I know what are characters in $\ell^1(Z)$ but don't know how to determine…
XYZ
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