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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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Compatible for matrices

We know that an $r \times n$ matrix $A=(\alpha_{jk})$ defines a linear operator from the vector space $X$ of all ordered $n$-tuples of numbers into the vector space $Y$ of all ordered $r$-tuples of numbers. Suppose that any norm $\|\cdot\|_1$ is…
mathsstudent
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Prove by induction $ (V^nx)(t) =\int_{0}^{t} \frac{(t-s)^{n-1}}{(n-1)!}x(s)ds $

Prove by induction that $ (V^nx)(t) =\int_{0}^{t} \frac{(t-s)^{n-1}}{(n-1)!}x(s)ds $ Use this result to solve for f the integral equation $f(t) = sin t + \int_{0}^{t} f(s)ds$ where V is the $Volterra$ $operator$ on $L^2(0,1)$ $$(Vx)(t) =…
Sarah
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Why does the norm topology makes all linear functions continuous?

Assuming we have a normed vector space V (assume infinite dimensional, as trivial if finite dimensional), then why does the norm topology make all linear functionals on V continuos? I can't see how this is true. As a linear functional on a normed…
user58514
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$\{T(x_n)\}$ is Cauchy in $Y$ whenever $\{x_n\}$ is Cauchy in $X$, we need to show $T$ is continuous.

$X,Y$ be complex norm linear space which are not necessarily complete, let $T:X\rightarrow Y$ be a linear map such that $\{T(x_n)\}$ is Cauchy in $Y$ whenever $\{x_n\}$ is Cauchy in $X$, we need to show $T$ is continuous. My Attempt: construct two…
Myshkin
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The norm and the spectrum of $B(L^p(X,\mu))$

Let $(X,\mu)$ be a $\sigma$-finite measure space.For $g\in L^{\infty}(X,\mu)$,let $M_g$ be the multiplication operator on $L^p(X,\mu)$ defined by $M_g f=gf$. I want to know what is the norm and the spectrum in $B(L^p(X,\mu))$ of $M_g$
89085731
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H$^2$ and $l^2$ are isomorphic

The Hardy space $H^2(\mathbb{D})$ is defined to be the space of all functions $f$ >holomorphic on the unit disk $\mathbb{D}$ with the norm $\lVert \cdot \rVert_H$ $\lVert f \rVert_H^2=\sup_{0
user1223
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why we show metrization on sequence spaces

I started studying functional analysis and I encountered an example of a metrizable space whose general picture I can't fully grasp. Let $X$ be the set of all sequences of real numbers. Let $x_n=\{{\xi_i^{(n)}}\}_{i=1}^{\infty}$ and let …
Nikola
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What is the weak star topology on $L(X,U')$?

The following is taken from Peter D. Lax book Functional Analysis. Exercise 5 page 165. Let $X,U$ be Banach spaces. I want to define the weak* topology on the set of bounded linear maps from $X$ to the dual of $U$ denoted $\mathcal{L}(X,U')$. Am i…
OgvRubin
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Weak convergence as composite of functional and bounded linear map

The given question is Let $X,U$ be Banach spaces and suppose that $U$ is reflexive, let $M:X\rightarrow U$ be a bounded linear map and suppose that $x_n$ converges weakly to $x$ in $X$ denoted $x_n\rightarrow_w x$. Show that $Mx_n\rightarrow_w Mx$…
OgvRubin
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Pythagorean Theorem in an Inner Product Space

This is a question about understanding a proof. Here is what my professor did. However, I do not know how he reached the final conclusion where he states that $$ \begin{align} \|f\|^2 - 2 \sum_{k=1}^n a_k c_k + \sum_{k=1}^n a_k^2 = \|f\|^2 -…
Fomalhaut
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Failure for a sequence to tend to the delta function does not imply the existence of a continuous function whose functional value tends to infinity.

In the book Functional Analysis by Peter D. Lax the following definition is given. A sequence $\{k_n\}$ of continuous functions on $[-1,1]$ tends to the $\delta$ function if $$lim_{n\rightarrow \infty}\int_{-1}^{1}f(t)k_n(t)dt = f(0)$$ for all…
OgvRubin
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Is this a basis of $L^p([0,1])$?

Do the set of functions: $\{1 , \frac{1}{x+1}, \frac{1}{(x+1)(x+2)}, \dots \}$ form a (non-orthonormal) Schauder basis of $L^p([0,1])$ for $1 \le p < \infty$? How can I prove or disprove this statement?
Television
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Pre-image of a dense subspace is dense

Let $T : H\rightarrow H$ be a continuous, injective, symmetric operator on a real Hilbert space $H$. Let $V$ be a subspace of $H$ such that $T(V)$ is dense in $H$. Must $V$ be dense in $H$?
user15464
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Boundness by a specific differential operator

Denote $\mathbb{T}$ the 1-torus and let $G : H^1(\mathbb{T}) \to H^1(\mathbb{T})$ be Frechet differentiable such that $G(0) = 0$ and $DG$ maps bounded sets in $H^1$ into bounded sets in $\mathcal{L}(H^1).$ Consider the sequence of operators…
A. PI
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Estimate for a specific differential operator

Denote $\mathbb{T}$ the 1-torus and let $G : H^1(\mathbb{T}) \to H^1(\mathbb{T})$ be Frechet differentiable such that $G(0) = 0$ and consider the operator $$L = \sqrt{1-\Delta}.$$ Can we prove that there exists $C>0$ such that $$\|LG(u)\|_{L^2}…
A. PI
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