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Metamath Proof Explorer

Theorem normf

Description: The norm function maps from Hilbert space to reals. (Contributed by NM, 6-Sep-2007) (Revised by Mario Carneiro, 15-Dec-2013) (New usage is discouraged.)

		Ref	Expression
	Assertion	normf	⊢ norm_ℎ : ℋ ⟶ ℝ

Proof

Step	Hyp	Ref	Expression
1		dfhnorm2	⊢ norm_ℎ = ( 𝑥 ∈ ℋ ↦ ( √ ‘ ( 𝑥 ·_ih 𝑥 ) ) )
2		hiidrcl	⊢ ( 𝑥 ∈ ℋ → ( 𝑥 ·_ih 𝑥 ) ∈ ℝ )
3		hiidge0	⊢ ( 𝑥 ∈ ℋ → 0 ≤ ( 𝑥 ·_ih 𝑥 ) )
4	2 3	resqrtcld	⊢ ( 𝑥 ∈ ℋ → ( √ ‘ ( 𝑥 ·_ih 𝑥 ) ) ∈ ℝ )
5	1 4	fmpti	⊢ norm_ℎ : ℋ ⟶ ℝ