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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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Embedding compact iff resolvent compact

Let $T\colon X\to X$ be a closed Operator on a Banach space $X$ and $\mu$ in the resolvent set. Then the following two Statements are equivalent: (a) The embedding $i\colon D_T\to X$ is compact. (b) The resolvent $R_T(\mu)\colon X\to X$ is…
Salamo
  • 1,094
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If $X^{\ast}$ separable there exists $(x_n) \subset S_E$ such that $x_n \rightharpoonup 0$.

Let $X$ an infinite dimensional vector space. Show that if $X^{\ast}$ is separable, then there exists a sequence $(x_n) \subset S_E$, the unit sphere in $X$, such that $x_n \rightharpoonup 0$ in the $\sigma(X,X^{\ast})$ topology. This is an exercise…
L.F. Cavenaghi
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weak* convergence in $\ell^\infty$ doesn't imply weak convergence.

I am trying to prove this statement but I have problems when dealing with the dual of $l^\infty$. I have found a characterization of the weak star convergence in terms of the boundedness of the norm of the sequence, but I don't know how to use it...…
Axel
  • 61
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Continuously invertible operators

Let X and Y be normed spaces and let operators $A,B\in L(X,Y)$ continuously invertible (exists $A^{−1},B^{−1}∈L(X,Y)$). Prove that if $$\Vert B−A\Vert\leq \frac{1}{2\Vert A^{−1}\Vert},$$ then $$\Vert B^{−1}−A^{−1}\Vert\leq2\Vert A^{−1}\Vert^2\Vert…
user442291
  • 51
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Sobolev space dense

Consider the Hilbert space $L^2([-L,L])$. Is the Sobolev space $H^2([-L,L])\subset L^2([-L,L])$ dense? (Maybe this can be seen from the reason why $L^2([-L,L])$ is a Hilbert space?)
mathfemi
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Is the closure of a symmetric operator symmetric?

Every symmetric operator $A$ can be closed to an operator. Are this closed extensions symmetric? Are there any conditions that make this true such as $Dom(A)$ is densly defined in some Hilbert space?
yess
  • 1,002
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Existence of a function representation

Let $f: [0,1]^2 \to R$ be an arbitrary continuous in both arguments and increasing in the first argument function, and let $h: [0,1]^2 \to [0,1]$ be some arbitrary function. Does $\forall f,h$ there exists $g: R^2 \to R$ such that…
Ilia Krasikov
  • 21
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spectrum of continuous Schrodinger operator

For $W:R \to R$ is a bounded function, consider the Schrodinger operator $$[H_{W}y](x)=-y''(x)+W(x)y(x), x \in R$$ where the domain is $D(H_{W})=\{y:R \to R| y \in AC_{loc}(R), y' \in AC_{loc}(R), y''\in L^{2}(R)\}$. How to show the spectrum…
Fengpeng Wang
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The Hyperplane $H=\left [ f =\alpha \right ]$ is closed iff $f$ is continuous

In proof of "The Hyperplane $H=\left [ f =\alpha \right ]$ is closed iff $f$ is continuous" we choose $x_0 \in H^c$ so that $f(x_0) \neq 0$,for example,$f(x_0)<\alpha$ then we prove $f(x) < \alpha$,$\forall x \in B(x,r) \subset H^c$ and $f$ is…
Desunkid
  • 1,231
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3 answers

Unbounded operator on $\mathcal{C}([0,1])$ with the norm $L_1$

I'm trying to do this problem but I don't know: Show that, considering the continuous functions on $\mathcal{C}([0, 1])$ as a subset of $L_1([0, 1])$, the linear functional on this subset $f\mapsto f(\frac{1}{2})$ is not bounded. I have tried to…
Orkidea
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Proof of a theorem about finite dimensional normed linear spaces

Recently I was reading a proof of the following proposition, Any two norms on a finite dimensional normed linear space are equivalent. To prove this result the author used the following result without proof, Result. Let $X$ be a finite…
user170039
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a linear map is open if and only if the image of the unit ball contains a ball around 0

Let T:X-->Y be a linear map. --> this direction follows by definition of an open set. For the other direction, this is how I've started attempting it: Let U be open in X. take an element in T(U), say T(u) where u is in U. We want to show T(u) is…
lkjhgfdsa
  • 591
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Prove that $R: W^{\perp} \to (V/W)^*, \ Rx^* = \tilde{x}^*$ is a well-defined.

Let $(V, \| \cdot \|)$ be a normed vector space and $M^{\perp}$ be the annihilator of $M$. If $W\subset V$ is a closed linear subspace. Prove that $R: W^{\perp} \to (V/W)^*, \ Rx^* = \tilde{x}^*$, where $\tilde{x}^*(x+ W) = x^*(x)$ is well defined.…
Olba12
  • 2,579
2
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1 answer

Is a metric space without Cauchy sequences complete?

A metric space $X$ is said to be complete if every Cauchy sequence in $X$ has a limit which is an element of $X$. If a metric space lacks Cauchy sequences, e.g. $\mathbb{N}\subset\mathbb{R}$, is it then complete?
Anna
  • 337
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1 answer

Partial derivate in Sobolev space

Let be $u\in W^{m,p}(\mathbb{R^n})$. Show that if a one partial derivate of $u$ is zero, then $u = 0$. How I can be able to start this proof?
user46060
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