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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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If $V$ is Volterra Operator, proof that $\|V^n\| = \frac{1}{(n-1)!}$

Define the Volterra Operator $V:C([0,1])\to C([0,1])$ given by $$ Vf(x) = \int_0^x f(t) \, dt, \quad f \in C([0,1]). $$ I'm asked to proof that $\|V^n\| = \frac{1}{(n-1)!}$. My attempt: First, I'm gonna proof that $V^n f (x) = \frac{1}{(n-1)!} \,…
user 242964
  • 1,898
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2 answers

Solving the following problem of convexity

Let $E$ be a complex Hilbert space. Let $A=(A_1,\cdots,A_d)\in \mathcal{L}(E)^d$. Consider \begin{eqnarray*} W_{max}(A) &=&\{\alpha\in \mathbb{C}^d:\;\exists\,(z_n)\subset E\;\;\hbox{such…
Student
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Is my proof for Schauder's theorem in non-Banach spaces correct?

Schauder's theorem states: Let $X, Y$ be Banach spaces, let $T \in B(X, Y)$ be a bounded linear operator. Then $T$ is compact $\iff$ $T'$ is compact, where $T' \in B(Y', X')$ is the dual operator. The $\implies$ direction is true if $X, Y$ are…
Vincent
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Steps in Proof of Convolution Theorem

I am reading a proof of the Convolution Theorem and don't understand this part: $$\int |f(z)|\int |g(z-x)| \, dx \, dz = \int|f(z)|\|g\|_1$$ Why does $\int |g(z-x)|dx = \|g\|_1$ ?
juaninf
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Why we define topology on vector space in functional analysis

It is said that functional analysis is just infinite dimensional version of linear algebra. However, I am quite puzzled by this statement since we are mainly doing analysis on it. Another question is that why we want to define topology on vector…
89085731
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How do we take the Frechet derivative here? A question concernig a paper on the Kuramoto model

In the paper "DYNAMICAL ASPECTS OF MEAN FIELD PLANE ROTATORS AND THE KURAMOTO MODEL" by L. Bertini, G. Giacomin, AND K. Pakdaman we read $$\partial_t…
Conrado Costa
  • 6,071
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convergence of argmax

Let $f$ and $g$ be two continuous functions on an interval $[a,b] \subset \mathbb{R}$. Define for every $k \in \mathbb{N}$ $f^k(x)=(1-\varepsilon_k)f(x)+\varepsilon_kg(x)$ in which $\varepsilon_k \rightarrow 0$ as $k \rightarrow \infty$. Suppose…
elnaz
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In the space of bounded sequences which converge to $0$ there is one such sequence with a minimum distance to $(1,1,1,...)$

In $\ell^{\infty}$, there is exactly one sequence in $c_0$ that has the minimum distance to $(1,1,1,...)$. I believe this statement is actually false. However, I am having trouble formulating two such sequences with minimal distance to disprove…
Dragonite
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Suppose that $H$ is an infinite dimensional normed vector space, is every infinite dimensional subspace of $H$ with a basis closed?

Suppose that $H$ is an infinite dimensional normed vector space, is every infinite dimensional subspace of $H$ with a basis closed? In finite dimension, this is correct, but in infinite dimension, is this right?
Keith
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Example when equality holds in Bessel Inequality

Example of $x\in l^2$ such that $\sum_{k=1}^{\infty}|\langle x,e_k\rangle|^2\leq \|x\|^2$ has strict inequality where $(e_k)$ is an orthonormal sequence in $l^2$. My thinking: I think it's not possible As $\|x\|\…
Abhinav Jha
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Does strong convergence to 0 implies convergence to zero on compact sets?

I've been struggling with this for a bit and was wondering if anyone can give me a hint: Suppose $\{ T_n\}_1^\infty\subset \mathcal{L}\{ X,Y\}$ is a sequence of bounded operators from a banach space $X$ to a normed space $Y$, such that $\forall x\in…
kneidell
  • 2,488
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Hahn-Banach and seminorms

Let $V$ be a vector space, $\phi \in V'$, and let $p_1,\dots,p_n$ be seminorms on $V$ s.t. $|\phi (x) | \leq \sum_{k=1}^{n} p_k(x)$ for all $x \in V$. Prove that there are $\phi_1, \dots \phi_n \in V'$ s.t $$ \phi = \sum_{k=1}^{n} \phi_k,…
Olba12
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Dual space - bounded functionals?

I'm taking a course on functional analysis at the moment and the lecturer has defined the dual space of a normed space $X$ to be the space of all bounded linear functionals and denoted it $X^*$. However if I look at other references (such as…
IAlreadyHaveAKey
  • 1,215
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Every Hamel basis has cardinality continuum $(c)$

I found the demonstration of this assertion in a book: "Let the cardinality of some Hamel basis be $\kappa$. We can easily calculate the cardinality of the generated vector space: it is $\aleph_0(\kappa+\kappa^2+\ldots)=\aleph_0\kappa = \kappa$ and…
ketherok
  • 133
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Prove a linear operator is continuous wrt weak topology iff it is continuous at $0$ wrt weak topology

Let $\mathbb E$ and $\mathbb F$ be two Banach spaces and $T:\mathbb E\to \mathbb F$ a linear operator. Use the definition of the weak topology to show that $T$ is continuous with respect $\mathbb E$ being equipped with the weak topology…
mint
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