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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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Linear subspaces of the linear space of continuous functions on the interval $[-1,1]$

I have a question from my functional analysis course I am a little confused with; here goes: Let $X = C[-1,1]$ be the linear space of continuous functions on the interval $[-1,1]$ over the field $\mathbb{R}$. Consider the subset $M$ $=$ {${f(x)…
katherinebarry
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Exercise 2.22 in Brezis' Functional Analysis

I am trying to solve the excercise 2.22 from Brezis' Functional Analysis. $2.22$ The purpose of this exercise is to construct an unbounded operator $A: D(A) \subset$ $E \rightarrow E$ that is densely defined, closed, and such that…
David Alexander
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Weak convergence and weak* convergence in $l^\infty$

I'm looking for a sequence that converges weakly* in $l^\infty$ but not weakly. In $l^1$ I can take $(e_n)_n$ with $e_n(k)=\delta_{k,n}$, but I don't know if there is a simple example in $l^\infty$. Thank you!
JN_2605
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$T^2=T$ implies $\dim(T(E))<\infty$

Let $T:E\to E$ be a compact operator verifying $T^2=T$. I have to prove that $\dim(T(E))<\infty$. I have proved that $T(E)$ is closed, but I don't know how to continue. Thank you!
JN_2605
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kernel of a linear functional and a subspace of codimension. one

I have a problem in understanding the proof a theorem/ lemma involving the subspace of codimension one and the kernel of a non-zero linear functional. I understand the first part of the proof but couldn't decipher the second part. Although it says…
Octagonal Monk
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Functional Analysis Proof

I'm sorry that the title is very general indeed. I'm looking for a theorem/corollary that uses all of the following four theorems/concepts in its course. This may be rather ambitious, but any ideas? I am not looking for the proof (just yet). BCT,…
larkhall
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Prove statement about Minkowski functional

I have this following to prove: Let $X$ be normed space and $C \subset X$ be an open, convex subset of $X$ with $0\in C$. Show that: $C= \{x \in X: p_C(x) < 1\}$ With $p_C$ defined as the Minkowski functional of $C$: $p_C : X \to [0,\infty]$, $x \to…
Hồng Ngọc Trương
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Question in Volterra operator

Consider $V:L^2([0,1],\mathbb{C}) \rightarrow L^2([0,1],\mathbb{C})$ defined by $$(Vu)(x)=\int_0^xu(t)dt$$ (i) Calculate the adjoint $V^*$. $$V^*=\int_0^x \bar{u(t)}\ \ dt$$ (ii) Suppose $u \in C([0,1];\mathbb{C})$. Show that…
Ahmed
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Why is a strongly continuous one-parameter semigroup called a $C_0$-semigroup?

Why is a strongly continuous one-parameter semigroup called a $C_0$-semigroup? Isn't $C_0$ the set of continuous functions that vanish at infinity? Thanks and regards!
Tim
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Spectral measure of a self-adjoint operator.

I'm dealing with a proof of a theorem that was left for the reader (not a news sadly). The proposition states the following : Let A be a self-adjoint operator and $\{\mu_n\}_{n=1}^{N}$ a family of spectral measures. Then…
smexyary
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Question about a "w.l.o.g." statement in proof of Uncertainty Principle?

I read a proof on the Uncertainty Principle (see below) and although the technical part itself is relatively straight forward I still do not understand a certain "w.l.o.g." statement in the proof. Uncertainty Principle : For $f\in L^2(\mathbb R)$…
Cubi73
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Deforming a linear map a little preserves surjectivity

Let $X$ be a Banach space, and $A: X\to X$ be a surjective linear map. Define $$\eta(A) = \{\lambda\in \mathbb{C}: A - \lambda I \text{ is surjective}\}$$ where $I: X\to X$ is the identity. Show that $\eta(A)$ is an open subset of $\mathbb{C}$.…
Van Latimer
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Comments about Riesz Lemma

Here is mentioned the Riesz Lemma and an important fact on the proof: Proof of Riesz Lemma My question is: Everything there is okay for any closed subspace. However, why can't we take $\theta = 1$? Is something related to the dimension of the…
Rub
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Convergence in distrubution sense and convergence of distribution to a distrubution.

The convergence in the test function is as follow : Let $\mathcal D(U)$ the space of test function. $(\varphi _n)$ to $\varphi $ in $\mathcal D(U)$ if there is a compact $K\subset U$ that contain all support of the $\varphi _k$'s and $\partial…
Pierre
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A condition for a bounded $T:X^* \rightarrow Y^*$ being a dual operator

So I am trying to learn some functional analsysis, but the weak star stuff really confuses me. I came across this one and am completely lost. Let $X$ and $Y$ be Banach spaces, and let $F:X^*\rightarrow Y^*$ be a bounded linear operator. Show there…
L. Castania
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