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Normed and Banach spaces
Vector spaces with a length; completeness; the classical sequence and function spaces.
A norm measures length; a Banach space is a normed space with no missing limits. ℓ^p, L^p and the continuous functions with the sup norm are the standard examples. Picture it: the unit balls for different norms — a diamond, a circle, a square. Think it: completeness is what lets you solve equations by successive approximation (the contraction mapping theorem).
Үйлдэл: sqrt(3^2 + 4^2)
Алхам алхмаар
- \sqrt{3^{2} + 4^{2}} = 5
Power: 3^2 = 9.
Хариулт
5
Symbols used here
\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
The non-negative number whose square (n-th power) is x.
\mathbf{v},\ \vec{v}
vector
A quantity with magnitude and direction; a column of numbers.
A quantity with magnitude and direction; a column of numbers.
\lambda
lambda (eigenvalue)
The factor by which an eigenvector is stretched: Av = λv.
The factor by which an eigenvector is stretched: Av = λv.
\mathbf{u} \cdot \mathbf{v},\ \|\mathbf{v}\|
dot product, norm
Σ u_i v_i; the length of v, √(v·v).
Σ u_i v_i; the length of v, √(v·v).
\sup,\ \inf
supremum, infimum
Least upper bound, greatest lower bound.
Least upper bound, greatest lower bound.
\|f\|_p,\ \langle f, g \rangle
p-norm, inner product
Length of a function; the generalised dot product.
Length of a function; the generalised dot product.
How to: Normed and Banach spaces
- Power: 3^2 = 9.
Questions people ask
What is a Hilbert space?
A vector space with an inner product (so lengths and angles make sense) that is complete (no missing limit points). Square-integrable functions form one; quantum states live in one.