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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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The application of the open mapping theorem

Let $V$ be a Hilbert space and $V_i \subset V(i=1, \ldots, J)$ a number of closed subspaces satisfying $V=\sum_{i=1}^J V_i$, which, by a simple application of the Open Mapping Theorem, implies $$ \sup _{\|v\|=1} \inf _{\sum_i v_i=v}…
Chandler
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bounded normal operator and spectrum

Problem: If A is a bounded normal operator, the spectrum $\sigma(A)=\{s+it:s \in \sigma(B),t \in \sigma(C)\}$, where B, C are bounded self adjoint operators which commute. Fact: A bounded normal operator A can be written $A=B+iC$, where B,C are…
neveryield
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If $T:X\to Y$ is continuous and $T^{t}:Y^{*}\to X^{*}$ is compact, is it true that $T$ is compact?

I have a question. I have Banach spaces $X$ and $Y$, and $Y$ is reflexive. If $T:X\to Y$ is continuous, and $T^{t}:Y^{*}\to X^{*}, T^{t}(\phi)=\phi \circ T$ is compact, is it true that $T$ is compact? Thanks so much
user89940
  • 388
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Non-existence of a continuation

Let $U \subset \ell^1$ with $$U = \{(x_n)_n \in \ell^1 | x_{2k−1} = 0, \text{ for all } k \in \Bbb{N}\}.$$ I need to show that with one exception (which one?) every continuous functional on $U$ admits infinitely many continuous linear continuations…
user1117620
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Find Riesz's representative of integration functional in Sobolev space $H_0^1(\Omega)$

Consider functional $$I_a^b(u) = \int_a^b u(t) dt$$ defined on the Sobolev space $H_0^1([c,d])$, where $c
Jan Olszewski
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Operator of differentiation

Okay I need to know once and for all how to properly show that $L$ is closed and not continuous, $L_n$ is bounded for all $n \in \mathbb{N}$, $L_n \to L$ pointwise when $n \to \infty$, $L_n$ doesn't converge to $L$ with respect to $\| \cdot…
user1127565
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why the spectrum of an operator with compact resolvent is discrete

I want a proof of this proposition: "The spectrum of an operator with compact resolvent is discrete"??? Thank you very much
mario
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What is the resolvent and spectrum of the projection operator?

What is the resolvent and spectrum of the projection operator? Well, i am reading about this subject and for one projection operator $P:X\to X$ ($P^2=P$) the excercise ask me to find the resolvent and spectrum. (The excersice do not say who is $X$.…
weymar andres
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What does "Hypermaximal operator" mean today?

I have heard some of my physics professors mention that hypermaximal means to have an infinite number of self-adjoint extensions. However, this was only mentioned during lectures and I could find no mention of this in math resources while scouting…
Mirod the best
  • 377
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Is the image of a closed subset in a Banach space under a lineal and continuous projection always closed? If the answer is no, is there any example?

I was wondering if the following conjecture is true: let $X$ be a Banach space, let $P: X\to X$ be a continuous linear projection, and let $C$ be a closed subspace (unrelated to $P$) of $X$. Then the image $P[C]$ is closed. By saying that $C$ is not…
Xavier González
  • 61
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Is there a closed set which doesn't have a minimal distance to a point?

I am currently studying functional analysis and in the lecture we had the following theorem: Let $X$ be a reflexive normed space. Let $A \subset X$ be a convex, closed non-empty subset and let $x_0 \notin A$ be a point. Then there is an $x \in A$…
Hannes Lehmann
  • 83
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How to decompose function into a product?

There is a function $f(x)$ obtained from fit to experimental data. The shape of the function is not fixed yet, the data can be fitted to various curves, e.g. to exponential or polynomial. I need to choose this function so that when $x$ is…
zeliboba
  • 357
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Interesting example of reintroducing a topology

Is there an interesting example of an infinite dimensional Topological Vector Space whose topology was initially induced by something other than a norm or inner product and was latter found out to be a normed/inner product space?
Alek Fröhlich
  • 521
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Krein Milman Theorem: Nonemptiness of extreme point

Conway says if $K$ is a nonempty compact convex subset of a locally convex Hausdorff space $X$, then ext$K \neq \emptyset$ and $K = \bar{\text{co}}(\text{ext}K)$. Here the compactness of $K$ and closed convex hull depends on the topology of $X$. But…
Jiya
  • 395
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Linear dependence of functionals (Intuition)

I'm trying to understand why the linear dependence theorem of functionals is true at an intuitive level. I know the proof given in Brezis's functional analysis book (lemma 3.2), where the Hahn-Banach theorem is used; despite all formal details of…
John Mars
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