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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 definition of convex hull: Is each element of a convex hull a convex combination of a finite number of elements?

In the definition of the convex hull of a set $A$, does it matter if each element of the convex hull is a convex combination of finite number of elements of $A$ or an infinite number? I am trying to prove that the convex hull of a bounded set is…
Teodorism
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Finding a sequence of linear functionals on $L^{\infty}[0,1]$ of norm 1 and converging to zero pointwise

I am totally stuck at the following problem... I need to find a sequence of linear functionals on $L^{\infty}[0,1]$ of norm 1 and converging to zero pointwise. That is, $\{f_n\}$ in the dual of $L^{\infty}[0,1]$ such that each $f_n$ is of norm 1 and…
Keith
  • 7,673
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Let $X$ be a normed space. If the closed unit ball is complete then $X$ is a Banach space.

I need to know how to prove this result. I do not have a great deal with normed spaces, so I have been trying to carry out this proof but I am not able to do it correctly. How could I do it?
user706948
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Why this spectrum problem is self-adjoint?

I am confused about why the following spectrum problem is self-adjoint: $\begin{matrix}y'(x) = \mathbf{M}(k,x)y(x)&(y:\mathbf{R}\rightarrow\mathbf{C}^2,x\in\mathbf{R},k\in\mathbf{C})\end{matrix}$ and the operator $\begin{matrix}\mathbf{M}(k,x) =…
Ansel
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Fourier Transform of quadratic Volterra series

Volterra functional series representation of nonlinear random processes \begin{align} y(t)&=h[x(t)]=y_1 (t)+y_2(t)+y_3…
MathArt
  • 1,053
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prove that specific linear transformation is bounded

this is the last question I was not able to answer to prepare for my final. Consider the linear transformation $T: \Bbb R^3 \to \Bbb R^2$ defined, for $x = (x_1, x_2, x_3) ∈ \Bbb R^3$ , by $$T(x) = (x_2-x_3 \,,\,x_1+2x_2)$$ where $\Bbb R^3$ and…
Omar
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How to show boundedness and calculate the norm of the linear transformation?

I am having a lot of trouble with this: $X$ denotes the Banach space $C[0,1]$ of continuous functions on $[0,1]$, with $\Vert f\Vert= \max\limits_{x\in [0,1]} |f(x)|$. Consider the linear transformation $T\colon X \to X,$ where $T(f)$ is a…
Omar
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Is a linear closed and everywhere defined operator bounded?

Let $X$ be a Banach space and let $A \colon X \to X$ be a linear operator. ($D(A) = X$) Prove of disprove that if $A$ is closed then it is necessarly bounded. (I'm having troubles in finding a connection between closedness and boundedness) Thank you…
user67133
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Given $\epsilon>0$, let $B_n^\epsilon=\{y\in Y:||y-y_n||<\epsilon||y_n||\}$. Then $\bigcup B_n^\epsilon\supset Y\backslash\{0\}$.

Let $Y$ be a separable Banach space. Let $\{y_n\}_1^\infty$ be a countable dense set in $Y$. Given $\epsilon>0$, let $B_n^\epsilon=\{y\in Y:||y-y_n||<\epsilon||y_n||\}$. Then $\bigcup B_n^\epsilon\supset Y\backslash\{0\}$. If…
Knt
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How do I show that the mapping of orthogonal complement of $\ker T$ to $\mathrm{range}\ T$ is one to one?

Having trouble figuring this out: Show that the mapping $T : (\ker T)^{\perp} \to \mathrm{range}\ T$ is one to one. I have the definitions but am I suppose to set them equal to each other?
User69127
  • 1,168
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Showing an operator is continuous

When $C[a,b]$ is the space of all real-valued continuous function defined on $[a,b]$ endowed with the uniform norm, for $F: C[a,b] \to C[a,b]$ given by $F(x)(t) = \int_a^t x(s)ds$ to show continuity can I argue that for all $t$ it holds that…
student24
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Composition of projections

If $P$ in $L(X,Y)$ and $Q$ in $L(Y,Z)$ are projections, Can we conclude that $QP$ is also a projection ? Thank you !
user60184
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1 answer

What function spaces are dense in $W^{k,p}(U)$? How to remember them?

Let $U\subset \mathbb{R}^n$ be a bounded open set. I know that, for example, $C^1(U)$ and $C_0^1(U)$ are dense in $W^{1,p}(U)$ and $W_0^{1,p}(U)$ respectively etc. Is it true that $C^\infty(U)$ is always dense in $W^{k,p}(U)$ for all $k,p$? Is it…
user66352
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$E\subset C[0,1]$ closed with functions of $C^1$ is finite dimensional

I want to prove that if $E$ is a closed linear subspace of $C[0,1]$ and the elements of $E$ are functions of class $C^1$, then $E$ is finite dimensional. I have tried to prove that the unit ball in $E$ is compact, but I don't know how to do it. Can…
JN_2605
  • 479
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2 answers

Compact operator $T:l^1\to c_0$

Given the operator $T:l^1\to c_0$ with $T(x=(x_k))=\left(\sum\limits_{k=1}^\infty x_k,\sum\limits_{k=2}^\infty x_k,...,\sum\limits_{k=n}^\infty x_k,...\right)$. I have to decide if this is a compact operator. I think it is, and to prove it I have…
JN_2605
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