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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 Mountain Pass theorem

I cam across the Mountain Pass Theorem, mentioned for example at http://en.wikipedia.org/wiki/Mountain_pass_theorem. In (very) loose terms, it somewhat reminds me of Rolle's theorem. Trying to understand it better in the infinte-dimensional…
An aedonist
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The relation between the (algebraic) dimensions of a normed linear space and its dual.

What is the relation between the (algebraic) dimensions of a normed linear space and its dual, for example can we say $\dim X \leq \dim X^*$, for a normed linear space $X$?
Arman
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Geometrical representation of the unit ball?

Let $E$ be the vector space of $\mathbb{R}$-valued continuous functions on $[0\ 1]$. With the norm $\| f \| = \max \{\ | f (x) |; 0 \leq x \leq 1\}$, the open ball centered at $f$ and radius $r$ has a simple graphical representation: it is a…
Piquito
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Equivalence of norms in Sobolev space

I am trying to prove an equivalence between two norms in the Sobolev space $H^1(\Omega)$ over a bounded Lipschitz domain $\Omega$, namely the standard norm $$||u||_{H^1(\Omega)}^2=\int_{\Omega} u^2 \,dx + \int_{\Omega} |\nabla u|^2\,dx$$ and the…
Luc
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I need help understanding the proof of Lemma 2.4-1 from Kreyszig's Functional Analysis.

Lemma: Let $\{x_1, \ldots, x_n \}$ be a linearly independent set of vectors in a normed space $X$ (of any dimension). Then there is a number $c > 0$ such that for every choice of scalars $\alpha_1, \ldots, \alpha_n$, we have $$\Vert \alpha_1 x_1 +…
banana.space
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what are differences between metric space and metric linear space?

A metric linear space is a linear space equipped with metric but i want to know the point wise differences between metric space and metric linear space.Can any body write it down in points?
Pushpa Shrestha
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Integration by parts for weak derivatives

I'm trying to show that if $g$ is such that $f(b) - f(a) = \int_a^b g(t) dt$ for any $a
Wooster
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Use Poisson summation formula to prove Gaussian sum formula

The Poisson summation formula states that for any Schwartz function $f$, $\sum\limits_{k\in\mathbb{Z}}f(k)=\sum\limits_{k\in\mathbb{Z}}\hat{f}(k)$, where $\hat{f}$ is the Fourier transform of $f$. The question that I am trying to solve asks me to…
Aden Dong
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Pointwise convergence resisting averaging

Can you give an example of a sequence of continuous functions $f_n:[0,1]\to [0,1]$, such that $f_n\to 0$ pointwise and there is no subsequence $(f_{n_k})$ for which $\frac 1 m\sum_{k=1}^{m}f_{n_k}$ tends to zero uniformly? I think it's the same as…
KotelKanim
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$c_0$ is not isometric to $c_0 \oplus c_0$

$c_0$ is the Banach space of sequences converging to zero and $c_0 \oplus c_0$ is its algebraical direct sum with itself equipped with the norm $\|(\xi,\eta)\| := \|\xi\|+\|\eta\|$. How to prove that this spaces is not linearly isometric (though…
quartermind
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Show that the Open Mapping Theorem requires both spaces to be complete

I am trying to show counter examples to the Open Mapping Theorem. In this particular case, I am trying to show that both spaces need to be Banach. First the OMT: Let $X, Y$ be Banach spaces. Let $T : X \rightarrow Y$ be a surjective bounded…
Tyler Hilton
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Minimal distance in a non-Hilbert space

It is well known that if $X$ is a Hilbert space and $W \subset X$ is a closed subspace, then for each $x \notin W$, there is a unique element of $W$ which lies at minimal distance from $x$. However, I could not think of an explicit example to show…
user15464
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limit of a sequence of functionals

Consider a sequence of functionals $(f_n)$, $f_n(x)=\int_{-1}^1x(t)\cos(nt)dt,\ n\geq 1$, on the space $L_2(-1,1)$. I need to prove that $f_n(x)\to 0$, as $n\to\infty$, for all $x\in L_2(-1,1)$. I know that $\int_{-1}^1\cos(nt)dt\to 0$, as…
nokiddn
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Derivative of Rayleigh quotient

I'm going over the proof of the spectral theorem for compact symmetric operators in Hilbert space in Lax. Let $A$ be a compact symmetric operator on a Hilbert space to itself. Define the Rayleigh quotient to be $$R_A(x) = \frac{(Ax,…
user901823
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Dual space of a finite dimensional normed space

My lecturer gave us this result today in class, but he didn't give a proof, he said we can prove it ourselves, only I'm really struggling to see how to do it. Let $E$ be a normed space with dual $E'$. Then $E$ is finite dimensional if and only if…
Victoria
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