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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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Explicit contradiction from assumption that $C([0,1], \Bbb R)$ is finite-dimensional

I, just for fun, want to find a contradiction to the assumption that $C([0,1],\Bbb R)$ is finite-dimensional by constructing a continuous function that cannot be a linear combination of the proposed finite Hamel basis $\{f_1, ..., f_N \}$. To be…
Christian Fieldhouse
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Proof of Kolmogorow-Riesz for $\ell^p$ space

I want to prove the following theorem: Let $1\le p < \infty$ and $K\subset \ell^p$. If $\forall n\in\mathbb{N},\ \exists M_n>0$, such that $\sup_{a\in K} |a_n| 0, \ \exists N(\varepsilon)$ with…
EpsilonDelta
  • 2,079
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Axioms of a metric

The following question is from Kreyszig, Introductory Functional Analysis. $(M2) d(x,y)=0$ iff $x=y.$ $(M3) d(x,y)=d(y,x)$. $(M4) d(x,y)\le d(x,z)+d(z,y)$. Show that (M3) and (M4) can be obtained from (M2) and $d(x,y)\le d(z,x)+d(z,y)$. If I…
jim
  • 804
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If $M\oplus N=B$ is a Banach space and $M$ is closed, does that imply that $N$ is closed as well?

I am aware a similar question had been answered before on the site, see(A question about complement of a closed subspace of a Banach space). I find the answer given there unsatisfying since it relies on a categorical definition of direct sum where…
roy yanai
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Are the even function closed in $C [-1,1]$?

We have a space $V=C ([-1,1]; \mathbb{R})$ with inner product: $$\langle f,g\rangle=\int_{-1}^1 f(x)g(x) dx.$$ Are the even function closed in $V$ in norm induced by inner product?
Kornel
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Existence of sequence with unity norm such that the distance between any two in norm sense is always greater than 1.

Let $X$ be infinite dimensional normed linear space. Then prove that there exist a sequence $\{x_n\}$ in $X$ such that $\|x_n\|=1$ and $\|x_n-x_m\|>1$ for all $n,m=1,2,...,n!=m$. Can this be solved by using the riesz lemma?
Ppp
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On basic functional analysis

Suppose $T$ is right shift operator on $l^{2}\mathbb(N)$, $A$ is bounded operator on $l^{2}\mathbb(N)$ and $||A-T||<1$, prove that $A$ is not invertible.
user548061
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1 answer

Corollary of Banach Steinhaus Theorem on compact set

I'm leanrning Banach Steinhaus Theorem and saw this on Wikipedia on one of the corollary of Banach Steinhaus Theorem: Since $\{Tn\}$ is bounded in operator norm, and the limit operator $T$ is continuous, a standard "3-ε" estimate shows that $T_n$…
nekodesu
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Extending real functional to complex functional? Do I need Hahn-Banach?

I'm trying to extend a real functional $f:V \rightarrow \mathbb{R}$ ($V$ is a complex vector space, boundedness of $f$ not known a priori) to the complex functional $f_{\mathbb{C}}: V \rightarrow \mathbb{C}$. If I need H-B, then how can I discover…
mavavilj
  • 7,270
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Norm of Bilinear Map Definition Confusion

I must be missing something simple... For a bilinear map $B: X \times Y \rightarrow Z$, the norm of the map is defined as $$\|B\| :=\sup_{\|x\| \leq 1, \hspace{1mm} \|y\| \leq 1} \|B(x, y)\|_Z.$$ This definition implies the following…
BenB
  • 837
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2 answers

$l^{\infty}$ is a Banach Space wrt sup norm (Functional Analysis)

$l^{\infty} := \{x=(x_n) : \sup_{n\in \mathbb N}|x_n| \lt \infty\} $ $\|x\|_{\infty} = \sup_{n \in \mathbb N} |x_n|$ Show that $l^{\infty}$ is a Banach Space (Complete) with respect to > $\|x\|_{\infty}$ If I wish a space is Banach, I must show…
usereb
  • 817
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is the sum of an increasing function and a decreasing function either monotonic or unimodal?

Suppose $f(x)$ is an increasing function and $g(x)$ a decreasing function, and $h(x)=f(x)+g(x)$, what guarantees $h'(X)$ has at most one solution? To be more specific, for the problem I have at hand, $f''(x)>0$ and $g''(x)<0$, is there at most 1…
Adam
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Normaloid operator

Let $B(H) $ be the Banach algebra of all bounded linear operators acting on an infinite dimensional complex Hilbert space $ H $. Let $ A, B\in B(H) $, such that $$ r(A)= \Vert A \Vert,$$ $$ r(B)= \Vert B \Vert .$$ Where $r(A)$ is the specral…
A. Bag
  • 141
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Calculating a norm of an operator defined in the continuous spaces: $||T||=||T(1)||_{\infty}$

If $T:C(X,\mathbb{R})\rightarrow C(Y,\mathbb{R})$ is an positive linear application, then $T$ is continuous and $||T||=||T(1)||_{\infty}$, where $1\in C(X,\mathbb{R})$ and $||f||_{\infty}=\max_{x\in X}|f|$. Now, let $\varphi…
Alex Pozo
  • 1,290
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Find the sum of the series $\frac{\sin(nx)}{n!}$

Find the sum of the series using the Fourier series $\sum_{n=1}^\infty \frac{\sin(nx)}{n!}$. I think I should find a function that in the expansion in a Fourier series gives something similar on the formula above.
user1223
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