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1233dfv
  • Member for 10 years, 8 months
  • Last seen more than 2 years ago
8 votes
3 answers
30k views

$\mathbb{C}$ is a one-dimensional complex vector space. What is its dimension when regarded as a vector space over $\mathbb{R}$?

7 votes
3 answers
764 views

Show that there exists a diagonal matrix $B$ the diagonal entries of which are $±1$ such that $A + B$ is nonsingular.

4 votes
6 answers
4k views

Show that every subspace of $\mathbb{R}^n$ is closed [closed]

4 votes
1 answer
1k views

$M$ and $N$ are subspaces of a Hilbert space. If $M\subset N$, show that $N^{\perp}\subset M^{\perp}$. Show also that $(M^{\perp})^{\perp}=M$.

4 votes
2 answers
2k views

Use Taylor's Theorem with $n=2$ to prove that the inequality $1+x<e^x$ is valid for all $x\in \mathbb{R}$ except $x=0$.

3 votes
1 answer
3k views

Confusion with parallel vector definition.

3 votes
4 answers
2k views

system of modular equations.

3 votes
1 answer
82 views

Let $V$ be an inner product space. Show that if $||x+y||=||x||+||y||$, then $ax=by$ where $a,b$ are non-negative and not both zero.

3 votes
2 answers
3k views

Triangle Inequality for norm integral $\|f\|_1=(\int_a^b [|f|^2+|f'|^2]\mathsf dx)^{1/2}$.

3 votes
2 answers
2k views

Upper and Lower Darboux integral of a piecewise function $f(x)=x$ and $f(x)=0$.

3 votes
1 answer
118 views

Find $\nabla_{\gamma'(t)}\gamma'(t)$. A metric on $\mathbb{R}^2$ is given by the form $dr^2+ f(r,\theta)d\theta ^{2}$ in polar coordinates. [closed]

3 votes
4 answers
4k views

Prove Newton's iteration will diverge for these functions, no matter what real starting point is selected. $f(x)=x^2+1$ and $g(x)=7x^4+3x^2+\pi$.

3 votes
3 answers
721 views

Find the Jordan canonical form of a $3\times 3$ antisymmetric matrix

2 votes
2 answers
129 views

Show that $\ln(1+x)\leq x-{1\over 2}x^2+{1\over 3}x^3$.

2 votes
2 answers
103 views

Find the limit of $\lim_{n\to \infty}n^2({1\over{n^3+1^3}}+{1\over{n^3+2^3}}+\cdots+{1\over{n^3+n^3}}).$

2 votes
2 answers
123 views

Show that there exists no positive continuous function $f$ defined on $[a,b]$ that satisfies the following conditions:

2 votes
2 answers
240 views

$||f||_1 =(\int_a^b [|f|^2+|f'|^2]dx)^{1/2}$. Is this normed space complete?

2 votes
2 answers
2k views

How do I show that in a normed space $| (\|x\|-\|y\|) | \leq \|x-y\|$ [duplicate]

2 votes
2 answers
602 views

Give an example of a non-self-adjoint operator on a Hilbert space $H$ whose range is $H$ and which is not invertible.

2 votes
2 answers
571 views

Compact linear operators $S$ and $T$, show that $S(I-T)=I$ if and only if $(I-T)S=I$ and deduce that $I-(I-T)^{-1}$ is a compact operator

2 votes
1 answer
71 views

Show by example that $AB=I$ does not imply that $BA=I$, with $I$ being the identity operator on $Y$. What is a suitable $Y$ for this to hold?

2 votes
1 answer
314 views

If $(\lambda_n)_{n=1}^\infty$ is a bounded sequence, then there is a bounded linear operator $A$ on a Hilbert space $H$ such that $Ae_n=\lambda_n e_n$

2 votes
1 answer
87 views

For $w\in\Bbb{C}$ with $|w|<1$, describe the set defined by $|z-w|\leq|1-\bar{w}z|$.

1 vote
2 answers
854 views

Find Taylor series for $f(x)=e^x$ at $c=3$. Then simplify the series and show how it could have been obtained directly from the series $f$ at $c=0$.

1 vote
1 answer
41 views

Find subsets $W$ and $V$ of $\mathbb{R}^3$ such that $\mathbb{R}(W\cap V)\neq\mathbb{R}W\cap \mathbb{R}V$.

1 vote
1 answer
3k views

For small values of $x$, the approximation $\sin(x)\approx x$ is often used. Estimate the error using this formula with the aid of Taylor's Theorem.

1 vote
2 answers
60 views

Let $F$ be a field. Find a matrix $A ∈ M_{4×4}(F)$ satisfying $A^4 = I \neq A^3$.

1 vote
1 answer
177 views

On the space $l_2$ we define an operator $T$ by $Tx=(x_1, {x_2\over2}, {x_3\over3}, . . . )$. Show that $T$ is bounded, and find its adjoint. [duplicate]

1 vote
1 answer
428 views

If a sequence of self-adjoint linear operators is convergent, show that its limit is self-adjoint.

1 vote
1 answer
563 views

inner product space problem $(x_n,y_n)\to 0$