A vector space $E$, generally over the field $\mathbb R$ or $\mathbb C$ with a map $\lVert \cdot\rVert\colon E\to \mathbb R_+$ satisfying some conditions.

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34 views

Equivalence of norms between two Banach spaces

I just learnt open mapping theorem. And I met a statement online asserting that If $X$,$Y$ are Banach space, and $T:X\to Y$ be a continuous bijection, then norms for $X,Y$ are equivalent. Can we ...
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1answer
26 views

A question involving norms

Let $(X, \| \cdot \|_X), (Y, \| \cdot \|_Y)$ be normed spaces and $T : X \rightarrow Y$ a bounded operator. Let $x, y \in X$ and let the norm on $X$ $$ \|x\| = \|x\|_X + \|Tx\|_Y. $$ I can't show that ...
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2answers
106 views

What are the best books for studying functional analysis in the world

I want to ask you maybe strange question but I really need answer What are the best books for studying functional analysis After Afew week I start study in master so I want references
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1answer
22 views

$f \in \mathcal{C}_{0}(X) \Rightarrow \sup_{x \in X} |f(x)| = \max_{x \in X} |f(x)|$

For a topological space $X$ we define $$\mathcal{C}_{0}(X) : = \left\lbrace f \colon X \longrightarrow \mathbb{C} \ \text{continuous} \colon \forall \, \varepsilon >0 \ \exists \, K \subseteq X \ ...
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1answer
46 views

Why if $T$ is not a bounded operator then exists $ (x_n) $ that converges to $ 0_{X} $ for which $ \| T(x_n) \| \geq n^2 $ for all $ n $?

Let $X$ and $Y$ be normed spaces. Suppose that $ T: X \to Y $ is a linear operator and assume that $T$ is not bounded. Why with these assumptions can I say that exists a sequence $ (x_{n})_{n \in ...
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1answer
46 views

$\|f\|_2\le\|f\|_4^{\frac{2}{3}}\|f\|_1^{\frac{1}{3}}$

I want to prove that $$ \|f\|_2\le\|f\|_4^{\frac{2}{3}}\|f\|_1^{\frac{1}{3}} $$ I proved it by Holder inequality. But this is an exercise under "Interpolation". So I guess it can be proved using ...
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23 views

Meaning of Normed Space

I have the following notations: 1.$(L^1\bigcap L^\infty)(0,A)$. 2.$L^\infty ((0,A)^2)$ 3.$L^\infty(Q)$ where Q =$(0,T) \times (0,A)$. Can someone explain to me what does the L norm represents, their ...
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2answers
26 views

For $X \in M(n,\mathbb R)$ , let $||X||:=\sqrt{Trace(AA^t)}$ , then $||AB|| \le ||A||\space||B|| , \forall A,B \in M(n,\mathbb R)$?

Let $M(n,\mathbb R)$ be the set of all square matrices of size $n$ with real entries . For $A \in M(n,\mathbb R)$ , let $||A||:=\sqrt{Trace(AA^t)}$ , then is it true that $A,B \in M(n.\mathbb R) ...
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0answers
31 views

Proving finite dimensional normed linear space is complete , without using equivalence of norms on finite dimensional vector spaces

Every finite dimensional normed linear space , over the field of real numbers , is complete . I know a proof of this result by using "every norm on a finite dimensional real vector space is equivalent ...
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2answers
36 views

Is the parallelogram equality satisfied in $l^1$?

I can't show that the parallelogram equality is satisfied / or is not satisfied in $l^1$. If $(v_n), (w_n) \in l^1$, then we have $$ || (v_n) + (w_n) ||_1^2 + || (v_n) - (w_n) ||_1^2 = || (v_n + ...
2
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1answer
40 views

Determining whats the Induced Metric

I have the normed space $({\rm Lip}([0,1]), \|\cdot\|)$, where ${\rm Lip}([0,1])$ is all Lipschitz functions from $[0,1]$ to $\Bbb R$, and $$\|f\|=|f(0)|+\sup_{0\le x,y\le ...
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1answer
183 views

In a real normed linear space if $||x||=||y||$ implies $\lim_{n \to \infty} ||x+ny||-||nx+y||=0$ , then the norm comes from an inner-product space?

$(V,\|\cdot|)$ be a real normed linear space such that $\|x\|=\|y\|$ implies $\lim\limits_{n\to\infty} \|x+ny\|-\|nx+y\|=0$, then is it true that the norm comes from an inner-product space ?
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1answer
33 views

A real vector space is an inner product space if every two dimensional subspace is an inner product space ?

Is it true that a vector space over the field of real numbers is an inner product space if every two dimensional subspace is an inner product space ? does it have anything to do with Neuman-Jordan's ...
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0answers
21 views

Looking for a simpler proof of Day's characterization of inner-product spaces and related things

I know the theorem that if $(V,||.||)$ is a real normed linear space such that the parallelogram identity $||x+y||^2+||x-y||^2=2(||x||^2+|y||^2)$ holds , then the norm comes from an inner-product ...
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1answer
41 views

Showing a Set is Dense in $C[a,b]$

Show that $C_0[a,b]:= \{f \in C[a,b] : f(a)=f(b)=0 \}$ is $\| \cdot \|_2$-dense in $C[a,b]$. Is it also $\| \cdot \|_{\infty}$-dense in $C[a,b]$? In the relevant chapter of my textbook, I am given ...
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2answers
21 views

$f,g \in [0,1] , f<g $ , when is $U:=\{h \in C[0,1]:f(t)<h(t)<g(t), \forall t \in [0,1] \}$ a ball in $C[0,1]$ with respect to the sup metric

Let $f,g:[0,1] \to \mathbb R$ be continuous functions such that $f(t)<g(t),\forall t \in [0,1]$ , then under what additional conditions on $f,g$ can we conclude that $U:=\{h \in ...
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2answers
13 views

A question on Matrix limits

Consider $M(n, \mathbb R)$ , the set of all real square matrices of size $n$ , as an NLS with norm $||A||:=\sqrt{\sum_{i=1}^n \sum_{j=1}^n a_{ij}^2}=Trace(AA^t)$ , then is it true that a sequence of ...
2
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1answer
43 views

A consequence of the open mapping theorem

We let $f$ be a bounded and surjective linear map from the Banach space $X$ onto the Banach space $Y$ and put $$ r_0=\inf\{r: f(B^X(0,r)\supset B^Y(0,1)\}. $$ Using the open mapping theorem, I have ...
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0answers
29 views

Equivalence of norms in $\mathbb R^n$

Does anyone know some proofs where we use fact of equivalence of norms(I think some L continuity use this fact). Any literature of proofs would be very useful. Thank's a lot.
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1answer
45 views

Multiplication Operator and Supremum Norm

Let $m\in C[a,b]$. Consider on $(C[a,b], \|\cdot \|_{\infty})$ the multiplication operator $A: C[a,b] \to C[a,b], \quad Af = mf$. Prove that $\|A\| = \|m\|_{\infty}$. In my book, we are given the ...
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1answer
47 views

Convexity of a certain function connected to the norm

Suppose that we are given two vectors $x,y$ in a normed space $X$. Can we prove in general that the function $$t\mapsto \|x-ty\|$$ is convex? It is certainly the case if the normed space has ...
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2answers
62 views

Convex set weakly closed if and only if strongly closed as well

I'm looking for a proof that given $(X\textbf{ } \|\cdot\|)$ normed space, $M \subset X$ convex set, $M$ is weakly closed if and only if it's strongly closed as well.
3
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1answer
15 views

Norm of a functional given by difference of values

How to calculate the norm of such functional? $$\phi : \mathcal{C} ([0,1]) \ni f \rightarrow 3 f (\frac{1}{2}) - 5 f (\frac{2}{3}) \in \mathbb{K} (= \mathbb{R}, \mathbb{C})$$ If we equip ...
4
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2answers
39 views

Normed separable space, linearly independent $X_0 \subset X, \ \overline{linX_0} = X $

Could you tell me how to prove that a normed space $X$ is separable $\iff$ there exists an at most countable set of linearly independent vectors $X_0 \subset X$ such that $ \ \text{lin} X_0$ is ...
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0answers
16 views

What is $\sum_{i=1}^m \frac{1}{a_i^Tx-b_i}$

Ask a dumb question: We all know the following: $\sum_{i=1}^m |a_i^Tx-b_i| = ||Ax-b||_1$ How about $\sum_{i=1}^m \frac{1}{a_i^Tx-b_i}$? I think it is definitely not $ ...
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0answers
28 views

Proof that second Frechet derivative is symmetric?

Is there a "nice" way to prove that the second Frechet derivative of a function between normed spaces is symmetric? Any proofs that I've managed to find seem quite messy and don't really give any ...
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1answer
68 views

Does isomorphism between normed space implies equivalence of norm?

Let $E$ be a vector space, $\|\cdot\|_1,\|\cdot\|_2$ be two norms on $E$, if $T:(E,\|\cdot\|_1)\to (E,\|\cdot\|_2)$ be an isomorphism (linear bijection, $T,T^{-1}$ are bouned), then does this imply ...
3
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2answers
51 views

Good books about differentiation in normed spaces?

Typical functional analysis books don't seem to cover this subject at all, so I'm looking for some good books that deal with differentiation in normed spaces(Gateaux/Frechet derivatives etc.). ...
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1answer
41 views

Diameter of an open ball in a normed space

This is probably a silly question, but I'm reading some class notes that have the following proposition: In general it's true that $\operatorname{diam}( B(x,r) ) \leq 2r $ but in a normed space ...
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0answers
27 views

Additive function on normed linear space are linear if continuous

Is it true: Suppose $T$ is an additive function on normed linear space $X$ to a normed linear space $Y$, i.e. $T(x+y) = T(x)+T(y)$ for all $x,y\in X$ and $T$ continuous at a point. Would $T$ be ...
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0answers
19 views

Does isomorphism of duals implies isomorphism? [duplicate]

Does isomorphism (or isometric) of $X^*$ and $Y^*$ for $X,\:Y$ normed spaces (or banach) implies isomorphism (or resp. isometric) of $X$ and $Y$? I know that the other way around is true, but I never ...
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1answer
33 views

is a normed space clopen or open?

i'm having a problem with the definition of open, closed and clopen sets. I have understood the basic definitions, but then the teacher today in class said that the normed space is limitless. A ...
2
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1answer
36 views

Linearly independent vectors in normed vector space

Considering the sequence {$\textbf{x}_k\}_{k=1}^{N} $ in a normed vector space $X$. Assuming a constant $\alpha>0$ such that $\alpha \sum_{k=1}^{N} |c_k|^2 \leq || \sum_{k=1}^{N} c_k \textbf{x}_k ...
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1answer
28 views

Is $C_c$ dense in $L_p$ for $0<p<1$?

Let $C_c$ be the set of compactly supported functions on $\mathbb{R}$ that are infinitely differentiable. Let $S$ be the set of Schwartz functions. It is well known that $C_c$ (hence $S$) is dense in ...
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4answers
51 views

Show that $\ell^1$ is not complete with a certain metric

For $x=(x_j)_{j\in\mathbb N}\in \ell^1$ let $$\|x\|=\sup_{n\in \mathbb N}\left \Vert \sum_{j=1}^{n}x_j\right\Vert$$ Show that $(\ell^1,\|\cdot\|)$ is a normed space, but it is not complete. The ...
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0answers
26 views

Different examples in L^p spaces

I was introduced to the concept of $L^p$ spaces for the first time, and has a lot of questions on what exactly they are. If I consider the measure space $(\mathbb{R},\mathscr{B}, Leb)$, for any $p, ...
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2answers
29 views

Continuity of the multiplication map $f\mapsto x^2 f(x)$ between normed spaces

Let $F:C[0,2]\to C[0,2]$ be the map defined by $(F(f))(x)=x^2f(x)$. Show that $F$ is continuous as a function from $(C[0,2],\|\cdot\|_{\sup})$ to $(C[0,2],\|\cdot\|_{2})$. I read this solution: ...
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1answer
34 views

Additive function and continuity at a point

Does continuity at a point and Additive function imply continuity at all other points in a normed linear space. Is there some result like there exist a in field such that f(x) = ax for all x in normed ...
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2answers
76 views

Normable topology determined by its restriction to a finite number of factors?

Is it generally true that all norms $\|\cdot\|$ on a finite product of normed spaces $E_1\times\dots\times E_n$ with $\|(0,\dots,0,x,0,\dots,0)\|=\|x\|_i$ where $\|\cdot\|_i$ denotes the norm on $E_i$ ...
2
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1answer
31 views

On the greatest norm element of weakly compact set

Let $X$ be a Banach space and $K\subset X$ be a nonempty weakly compact set. I would like to know if there exists a point $u_0\in K$ such that $\|u_0\|\geq \|u\|$ for all $u\in K$. Thank you for all ...
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1answer
23 views

>>A. $X$ banch. >>B.M banach. >>C. $X/M$ banach.

I want to prove that any two will imply the other: A. $X$ banch. B.M banach. C. $X/M$ banach. I have proved A+B imply C by taking M closed in X. I am not understanding ...
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1answer
25 views

Show that the application $ Id:( C^1(0,1), \|\cdot\|_{1})\to ( C^1(0,1), \|\cdot\|_{\infty})$ is not continous

I want to prove that the application $$ Id:( C^1(0,1), \|\cdot\|_{1})\to ( C^1(0,1), \|\cdot\|_{\infty}) $$ is not continous. If I prove that this application is not bounded I have finished. So I ...
2
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0answers
48 views

n- dimensional normed linear space isomorphic to n-dimensional Euclidean space

If $(X,\|\cdot\|)$ is an n- dimensional normed linear space over R. Is it isomorphic to n-dimensional Euclidean space $R^n$. I know it is topologically isomorphic but what about isometry? I think if ...
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0answers
33 views

Covering Number of Lipschitz Function Space

On page 12 of the slide deck here, the author gives an example where a lower bound (and upper bound, but I am particularly interested in the lower bound) on the $\epsilon$-covering number of a ...
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1answer
43 views

Computing $\sup_{\left \| u \right \|=1} d(u,F)$ in a closed subspace of a normed space.

I have come across with the following problem: Let $E$ be a normed space and $F$ a closed subspace of $E$. It's asking to compute $\sup\limits_{\left \| u \right \|=1} d(u,F)$. What I know it's ...
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1answer
21 views

Does a normed space with norm2 defines an inner product?

I know that generally, an inner product defines a norm on an inner product space, But, generally speaking, If I have a normed space (on purpose I do not say which) with the norm 2 does it mean that I ...
1
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1answer
43 views

Bounded functional which composed with an unbounded operator becomes unbounded

While working on a problem, I came up with a certain lemma, however I'm not sure whether it's true and I'd be grateful for some insight. Let $ X $ and $ Y $ be normed spaces over reals, where $ X $ ...
0
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1answer
46 views

Isometry under condition $\|x-y\|=n\iff\|f(x)-f(y)\|=n.$ for $n\in\mathbf{N}$

Let $X, Y$ be normed spaces and $f:X\to Y$ be mapping and $n\in\mathbf{N}$ If$$\|x-y\|=n\iff\|f(x)-f(y)\|=n.$$ Under what conditions this map will be an isometry? Thanks
2
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2answers
63 views

Prove the triangle inequality is valid for the norm $\lVert f \rVert =(\int_0^1 \lvert f(x) \rvert ^2 dx)^{1/2}$

I.e, prove $\lVert f+g \rVert\ \le \lVert f \rVert + \lVert g \rVert$ for all $f,g$ in $C^\infty [0,1]$, $$\lVert f \rVert =(\int_0^1 \lvert f(x) \rvert ^2 dx)^{1/2}$$ I think we're supposed to use ...
-1
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1answer
58 views

Prove that L 2 PC[−1, 1] is not a complete normed space

I'm trying to prove that the normed space of all piecewise continuously functions with the norm $$\int^1_{-1}|f(x)|^2dx$$ is not a complete normed space. $L_2PC[-1,1]$ for that, im trying to find a ...